Search

We consider the problem of communicating over a channel that randomly “tears” the message block into small pieces of different sizes and shuffles them. For the binary torn-paper channel with block length $n$ and pieces of length ${\mathrm{ Geometric}}(p_{n})$ , we characterize the capacity as $C = e^{-\alpha }$ , where $\alpha = \lim _{n\to \infty } p_{n} \log n$. Our results show that the case of ${\mathrm{ Geometric}}(p_{n})$ -length fragments and the case of deterministic length- $(1/p_{n})$ fragments are qualitatively different and, surprisingly, the capacity of the former is larger. Intuitively, this is due to the fact that, in the random fragments case, large fragments are sometimes observed, which boosts the capacity. [ABSTRACT FROM AUTHOR]

Linear codes achieving or nearly achieving the Singleton bound are interesting in both theory and practice. The objective of this paper is to construct several infinite families of MDS, near MDS and almost MDS codes from some special cyclic subgroups of ${\mathbb {F}}_{q^{2}}^{*}$. To this end, the augmentation and extension techniques are used. The codes in this paper have flexible parameters and their lengths could be large. The minimum linear locality of the codes constructed in this paper is also studied. Some infinite families of optimal linearly locally recoverable codes are obtained. Besides, some codes in this paper are proved to be proper for error detection. [ABSTRACT FROM AUTHOR]

The number-theoretic code is a class of codes defined by single or multiple congruences. These codes are mainly used for correcting insertion and deletion errors, and for correcting asymmetric errors. This paper presents a formula for a generalization of the complete weight enumerator for the number-theoretic codes. This formula allows us to derive the weight enumerators and cardinalities for the number-theoretic codes. As a special case, this paper provides the Hamming weight enumerators and cardinalities of the non-binary Tenengolts’ codes, correcting single insertion or deletion. Moreover, we show that the formula deduces the MacWilliams identity for the linear codes over the ring of integers modulo $r$. [ABSTRACT FROM AUTHOR]

In this paper, we consider variable-length coding over the memoryless binary asymmetric channel (BAC) with full noiseless feedback, including the binary symmetric channel (BSC) as a special case. In 2012, Naghshvar et al. introduced a coding scheme, which we refer to as the small-enough-difference (SED) coding scheme. For symmetric binary-input channels, the deterministic variable-length feedback (VLF) code constructed with the SED coding scheme asymptotically achieves both capacity and Burnashev’s optimal error exponent. Building on the work of Naghshvar et al., this paper extends the SED coding scheme to the BAC and develops a non-asymptotic VLF achievability bound that is shown to achieve both capacity and the optimal error exponent. For the specific case of the BSC, we develop an additional non-asymptotic VLF achievability bound using a two-phase analysis that leverages both a submartingale synthesis and a Markov chain time of first passage analysis. Numerical evaluations show that both new VLF achievability bounds outperform Polyanskiy’s achievability bound for variable-length stop-feedback codes. [ABSTRACT FROM AUTHOR]

A functional $k$ -batch code of dimension $s$ consists of $n$ servers storing linear combinations of $s$ linearly independent information bits. Any multiset request of size $k$ of linear combinations (or requests) of the information bits can be recovered by $k$ disjoint subsets of the servers. The goal under this paradigm is to find the minimum number of servers for given values of $s$ and $k$. A recent conjecture states that for any $k=2^{s-1}$ requests the optimal solution requires $2^{s}-1$ servers. This conjecture is verified for $s \leqslant 5$ but previous work could only show that codes with $n=2^{s}-1$ servers can support a solution for $k=2^{s-2} + 2^{s-4} + \left \lfloor{ \frac { 2^{s/2}}{\sqrt {24}} }\right \rfloor $ requests. This paper reduces this gap and shows the existence of codes for $k=\lfloor \frac {5}{6}2^{s-1} \rfloor - s$ requests with the same number of servers. Another construction in the paper provides a code with $n=2^{s+1}-2$ servers and $k=2^{s}$ requests, which is an optimal result. These constructions are mainly based on extended Simplex codes and equivalently provide constructions for parallel Random I/O (RIO) codes. [ABSTRACT FROM AUTHOR]

In 1991, Wei proved a duality theorem that established an interesting connection between the generalized Hamming weights of a linear code and those of its dual code. Wei’s duality theorem has since been extensively studied from different perspectives and extended to other settings. In this paper, we re-examine Wei’s duality theorem and its various extensions, henceforth referred to as Wei-type duality theorems, from a new Galois connection perspective. Our approach is based on the observation that the generalized Hamming weights and the dimension/length profiles of a linear code form a Galois connection. The central result of this paper is a general Wei-type duality theorem for two Galois connections between finite subsets of $\mathbb {Z}$ , from which all the known Wei-type duality theorems can be recovered. As corollaries of our central result, we prove new Wei-type duality theorems for $w$ -demi-matroids defined over finite sets and $w$ -demi-polymatroids defined over modules with a composition series, which further allows us to unify and generalize all the known Wei-type duality theorems established for codes endowed with various metrics. [ABSTRACT FROM AUTHOR]

For fixed integers $n$ and $b\geq k$ , let $A(b,k,n)$ the largest size of a subset of $\{1,2,\ldots,b\}^{n}$ such that, for any $k$ distinct elements in the set, there is a coordinate where they all differ. Bounding $A(b,k,n)$ is a problem of relevant interest in information theory and computer science, relating to the zero-error capacity with list decoding and to the study of $(b, k)$ -hash families of functions. It is known that, for fixed $b$ and $k$ , $A(b,k,n)$ grows exponentially in $n$. In this paper, we determine new exponential upper bounds for different values of $b$ and $k$. A first bound on $A(b,k,n)$ for general $b$ and $k$ was derived by Fredman and Komlós in the ’80s and improved for certain $b\neq k$ by Körner and Marton and by Arikan. Only very recently better bounds were derived for general $b$ and $k$ by Guruswami and Riazanov, while stronger results for small values of $b=k$ were obtained by Arikan, by Dalai, Guruswami and Radhakrishnan, and by Costa and Dalai. In this paper, we strengthen the bounds for some specific values of $b$ and $k$. Our contribution is a new computational method for obtaining upper bounds on the values of a quadratic form defined over discrete probability distributions in arbitrary dimensions, which emerged as a central ingredient in recent works. The proposed method reduces an infinite-dimensional problem to a finite one, which we manage to further simplify by means of a series of optimality conditions. [ABSTRACT FROM AUTHOR]

A common method for reading information stored in DNA molecules is shotgun sequencing. This method outputs a histogram of the frequencies of all the molecules’ substrings of a given length $\ell $. To protect against noisy readings, the rank-modulation scheme encodes the information in the relative ranking of the substring frequencies, instead of their absolute values. However, the best rank-modulation codes for shotgun sequencing have low rates which are asymptotically vanishing. In this paper we propose new constructions of rank-modulation codes for shotgun sequencing. The first code construction is systematic, allowing the user to arbitrarily set the frequencies of a large subset of the substrings, which the encoder then completes to a permutation that may be realized by a DNA molecule. The construction is then improved by allowing the user to set the frequencies of additional substrings, at the cost of imposing constraints on the frequencies. The resulting codes have higher, non-vanishing rates, compared with previously known codes. As an example, for histograms of substrings of length $\ell =2$ , and an alphabet of size 4 (as in DNA molecules), we are able to construct a code with rate $\approx 0.909$ , whereas previously, the best construction resulted in a code with rate $\approx 0.654$. Additionally, the encoded information in our construction may be written to shorter DNA molecules than possible before. We also prove that the systematic codes constructed in this paper are the largest possible among all systematic codes. [ABSTRACT FROM AUTHOR]

Recently, subfield codes of geometric codes over large finite fields ${\mathrm {GF}}(q)$ with dimension 3 and 4 were studied and distance-optimal subfield codes over ${\mathrm {GF}}(p)$ were obtained, where $q=p^{m}$. The key idea for obtaining very good subfield codes over small fields is to choose very good linear codes over an extension field with small dimension. This paper first presents a general construction of $[q+1, 2, q]$ MDS codes over ${\mathrm {GF}}(q)$ , and then studies the subfield codes over ${\mathrm {GF}}(p)$ of some of the $[q+1, 2,q]$ MDS codes over ${\mathrm {GF}}(q)$. Two families of dimension-optimal codes over ${\mathrm {GF}}(p)$ are obtained, and several families of nearly optimal codes over ${\mathrm {GF}}(p)$ are produced. Several open problems are also proposed in this paper. [ABSTRACT FROM AUTHOR]

Linear codes generated by component functions of perfect nonlinear (PN for short) and almost perfect nonlinear (APN for short) functions and the first-order Reed-Muller codes have been an object of intensive study in coding theory. The objective of this paper is to investigate some binary shortened codes of two families of linear codes from APN functions and some $p$ -ary shortened codes associated with PN functions. The weight distributions of these shortened codes and the parameters of their duals are determined. The parameters of these binary codes and $p$ -ary codes are flexible. Many of the codes presented in this paper are optimal or almost optimal. The results of this paper show that the shortening technique is very promising for constructing good codes. [ABSTRACT FROM AUTHOR]

Linear codes with few weights have applications in consumer electronics, communications, data storage systems, secret sharing, authentication codes, and association schemes. As a special class of linear codes, minimal linear codes have important applications in secret sharing and secure computation of data between two parties. The construction of minimal linear codes with new and desirable parameters is an interesting research topic in coding theory and cryptography. Recently, Mesnager et.al. stated that “constructing linear codes with good parameters from non-weakly regular bent functions is an interesting problem.” The goal of this paper is to construct linear codes with two or three weights from non-weakly regular bent functions over finite fields and analyze the minimality of the constructed linear codes. In doing so, we draw inspiration from a paper by Mesnager in which she constructed linear codes with small weights from weakly regular bent functions based on a generic construction method. First, we recall the definitions of the subsets $B_{+}(f)$ and $B_{-}(f)$ associated with a non-weakly regular bent function $f$. Next, we construct two- or three-weight linear $p$ -ary codes on these sets using duals of the non-weakly regular bent functions that are also bent. We note that the constructed linear codes are minimal in almost all cases. Moreover, when $f$ is a non-weakly regular bent function in a certain subclass of Generalized Maiorana-McFarland bent functions, we determine the weight distributions of the corresponding linear codes. As far as we know, the construction of linear codes from non-weakly regular bent functions over finite fields is first studied in the literature by the second author in his dissertation. [ABSTRACT FROM AUTHOR]

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field ${\mathrm {GF}}(q)$ with length $q+1$ , which are closely related to the action of the projective general linear group of degree two on the projective line. Despite its interest, not much is known about this class of BCH codes. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$ -ary BCH codes with length $q+1$ , and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of ${\mathrm {PGL}}(2, p^{m})$ -invariant codes over ${\mathrm {GF}}(p^{h})$ is completed. As an application of this result, the $p$ -ranks of all incidence structures invariant under the projective general linear group ${\mathrm {PGL}}(2, p^{m})$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms. [ABSTRACT FROM AUTHOR]

It is well-known that cyclic codes have efficient encoding and decoding algorithms. In recent years, antiprimitive BCH codes have attracted a lot of attention. The objective of this paper is to study BCH codes of this type over finite fields and analyse their parameters. Some lower bounds on the minimum distance of antiprimitive BCH codes are given. The BCH codes presented in this paper have good parameters in general, containing many optimal linear codes. In particular, two open problems about the minimum distance of BCH codes of this type are partially solved in this paper. [ABSTRACT FROM AUTHOR]

This paper derives the optimal test channel distribution and the complete characterization of the classical Gorbunov and Pinsker (1973), Gorbunov and Pinsker (1974) nonanticipatory epsilon entropy of multivariate Gaussian Markov sources with square-error fidelity, which remained an open problem since 1974. The paper also formulates a state dependent nonanticipatory epsilon entropy, in which past reproductions are available to the decoder and not to the encoder, the test channel is specified with respect to an auxiliary (state) random process, and the reproduction process is a causal function of past reproduction and the auxiliary random process. This variation is analogous to the Wyner and Ziv (1976) and Wyner (1978) rate distortion function (RDF), of memoryless sources. It is shown that the operational rate of zero-delay codes, with past reproductions available to the decoder but not to the encoder is bounded below by the state dependent nonanticipatory epsilon entropy rate. For the case of multivariate Gaussian Markov sources with square-error fidelity, the optimal test channel distribution and the complete characterization of the state dependent of nonanticipatory epsilon entropy are derived, and also shown that that the two nonanticipatory epsilon entropies coincide. The derivations are new; they are based on structural properties of the stochastic realizations of the reproduction process that induce the optimal test channel distributions. They are derived using, achievable lower bounds on information theoretic measures, properties of mean-square estimation theory, Hadamard’s inequality, and canonical correlation coefficients of a tuple of multivariate jointly Gaussian random processes. Applications of the nonanticipatory epsilon entropy and its state dependent variation are discussed to the areas of control of unstable Gaussian systems over limited memory channels, design of causal estimators for Gaussian Markov sources with a fidelity criterion, computation of the rate loss of causal and zero-delay codes of Gaussian Markov sources with respect to non-causal codes. [ABSTRACT FROM AUTHOR]

Caching is an efficient way to reduce network traffic congestion during peak hours by storing some content at the users’ local caches. For the shared-link network with end-user-caches, Maddah-Ali and Niesen proposed a two-phase coded caching strategy. In practice, users may communicate with the server through intermediate relays. This paper studies the tradeoff between the memory size ${\mathsf M}$ and the network load ${\mathsf R}$ for the networks where a server with ${\mathsf N}$ files is connected to ${\mathsf H}$ relays (without caches), which in turn are connected to ${\mathsf K}$ users equipped with caches of ${\mathsf M}$ files. When each user is connected to a different subset of ${\mathsf r}$ relays, i.e., ${\mathsf K}= \binom {\mathsf H}{\mathsf r}$ , the system is referred to as a combination network with end-user-caches. In this work, converse bounds are derived for the practically motivated case of uncoded cache contents, that is, bits of the various files are directly pushed into the user caches without any coding. In this case, once the cache contents and the users’ demands are known, the problem reduces to a general index coding problem. This paper shows that relying on a well-known “acyclic index coding converse bound” results in converse bounds that are not tight for combination networks with end-user-caches. A novel converse bound that leverages the network topology is proposed, which is the tightest converse bound known to date. As a result of independent interest, an inequality that generalizes the well-known sub-modularity of entropy is derived. Several novel caching schemes are proposed, based on the Maddah-Ali and Niesen cache placement. These schemes leverage the structure of the combination network or/and perform interference elimination at the end-users. The proposed schemes are proved: (i) to be (order) optimal for some $({\mathsf N}, {\mathsf M}, {\mathsf H}, {\mathsf r})$ parameters regimes under the constraint of uncoded cache placement, and (ii) to outperform the state-of-the-art schemes in numerical evaluations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of generalized pair weights of an $[n, k]$ -linear code over the finite field $\mathbb {F}_{q}$ and the notion of pair $r$ -equiweight codes, where $1\le r\le k-1$. Some basic properties of generalized pair weights of linear codes over finite fields are derived. We obtain a necessary and sufficient condition for an $[n,k]$ -linear code to be a pair equiweight code, and we characterize pair $r$ -equiweight codes for any $1\le r\le k-1$. A necessary and sufficient condition for a linear isomorphism to preserve pair weights between two linear codes is obtained. At the end of this paper, an application of generalized pair weights of linear codes to symbol-pair read wire-tap channels of type II is introduced. [ABSTRACT FROM AUTHOR]

In general coding theory, we often assume that error is observed in transferring or storing encoded symbols, while the process of encoding itself is error-free. Motivated by recent applications of coding theory, in this paper, we consider the case where the process of encoding is distributed and prone to error. We introduce the problem of distributed encoding, comprised of a set of $K \in \mathbb {N}$ isolated source nodes and $N \in \mathbb {N}$ encoding nodes. Each source node has one symbol from a finite field, which is sent to each of the encoding nodes. Each encoding node stores an encoded symbol from the same field, as a function of the received symbols. However, some of the source nodes are controlled by the adversary and may send different symbols to different encoding nodes. Depending on the number of the adversarial nodes, denoted by $\beta \in \mathbb {N}$ , and the cardinality of the set of symbols that each one generates, denoted by $v \in \mathbb {N}$ , the process of decoding from the encoded symbols could be impossible. Assume that a decoder connects to an arbitrary subset of $t \in \mathbb {N}$ encoding nodes and wants to decode the symbols of the honest nodes correctly, without necessarily identifying the sets of honest and adversarial nodes. An important characteristic of a distributed encoding system is $t^{*} \in \mathbb {N}$ , the minimum of such $t$ , which is a function of $K$ , $N$ , $\beta $ , and $v$. In this paper, we study the distributed linear encoding system, i.e. one in which the encoding nodes use linear coding. We show that $t^{*}_{\textrm {Linear}}=K+2\beta (v-1)$ , if $N\ge K+2\beta (v-1)$ , and $t^{*}_{\textrm {Linear}}=N$ , if $N\le K+2\beta (v-1)$. In order to achieve $t^{*}_{\textrm {Linear}}$ , we use random linear coding and show that in any feasible solution that the decoder finds, the messages of the honest nodes are decoded correctly. In order to prove the converse of the fundamental limit, we show that when the adversary behaves in a particular way, it can always confuse the decoder between two feasible solutions that differ in the message of at least one honest node. [ABSTRACT FROM AUTHOR]

This paper studies the problem of constructing codes correcting deletions in arrays. Under this model, it is assumed that an $n \times n$ array can experience deletions of rows and columns. These deletion errors are referred to as $({t_{\mathrm {r}}}, {t_{\mathrm {c}}})$ -criss-cross deletions if ${t_{\mathrm {r}}}$ rows and ${t_{\mathrm {c}}}$ columns are deleted, while a code correcting these deletion patterns is called a $({t_{\mathrm {r}}}, {t_{\mathrm {c}}})$ -criss-cross deletion correction code. The definitions for criss-cross insertions are similar. It is first shown that when $t_{r}=t_{c}$ the problems of correcting criss-cross deletions and criss-cross insertions are equivalent. The focus of this paper lies on the case of (1, 1)-criss-cross deletions. A non-asymptotic upper bound on the cardinality of (1, 1)-criss-cross deletion correction codes is shown which assures that the redundancy is at least $2n-3+2\log n$ bits. A code construction with an existential encoding and an explicit decoding algorithm is presented. The redundancy of the construction is at most $2n+4 \log n + 7 +2 \log e$. A construction with explicit encoder and decoder is presented. The explicit encoder adds an extra $5\log n + 5$ bits of redundancy to the construction. [ABSTRACT FROM AUTHOR]

In this work, we consider the leakage-resilience of algebraic-geometric (AG for short) codes based ramp secret sharing schemes extending the analysis on the leakage-resilience of linear threshold secret sharing schemes over prime fields that is done by Benhamouda et al. in the effort to construct linear leakage-resilient secret sharing schemes with constant share size. Since there does not exist any explicit efficient construction of AG codes over prime fields with constant field size, we consider constructions over prime fields with the help of concatenation method and constructions of codes over field extensions. Extending the Fourier analysis done by Benhamouda et al., one can show that concatenated algebraic geometric codes over prime fields do produce some nice leakage-resilient secret sharing schemes. One natural and curious question is whether AG codes over extension fields produce better leakage-resilient secret sharing schemes than the construction based on concatenated AG codes. Such construction provides several advantage compared to the construction over prime fields using concatenation method. It is clear that AG codes over extension fields give secret sharing schemes with a smaller reconstruction threshold for a fixed privacy parameter $t$. In this work, it is also confirmed that indeed AG codes over extension fields have stronger leakage-resilience under some reasonable assumptions. Furthermore, we also show that AG codes over extension fields may provide strong multiplicative property which may be used in its application to the study of multiparty computation. In contrast, the same cannot be said for constructions based on concatenated AG codes, even when we are considering multiplication friendly embeddings. These advantages strongly motivate the study of secret sharing schemes from AG codes over extension fields. The current paper has two main contributions: (i) we obtain leakage-resilient secret sharing schemes with constant share sizes and unbounded numbers of players. Some of the schemes constructed without the use of concatenation also possesses strong multiplicative property (ii) via Fourier Analysis, we analyze the leakage-resilience of secret sharing schemes from codes over extension fields. This is of its own theoretical interest independent of its application to secret sharing schemes from algebraic geometric codes over extension fields. [ABSTRACT FROM AUTHOR]

Twisted generalized Reed-Solomon (TGRS) codes are a family of codes that contains a large number of maximum distance separable (MDS) codes that are non-equivalent to generalized Reed-Solomon (GRS) codes. In this paper, we characterize a sufficient and necessary condition that a twisted Reed-Solomon (TRS) code with two twists is MDS; give a sufficient and necessary condition that a TGRS code with two twists is self-dual, and present some constructions of self-dual TGRS codes. These self-dual codes are MDS, NMDS or 2-MDS. Furthermore, we study the non-GRS properties of TGRS codes with two twists and prove that these codes are non-GRS in most cases. [ABSTRACT FROM AUTHOR]

We study the list decodability of different ensembles of codes over the real alphabet under the assumption of an omniscient adversary. It is a well-known result that when the source and the adversary have power constraints $P $ and $N $ respectively, the list decoding capacity is equal to $\frac {1}{2}\log \frac {P}{N}$. Random spherical codes achieve constant list sizes, and the goal of the present paper is to obtain a better understanding of the smallest achievable list size as a function of the gap to capacity. We show a reduction from arbitrary codes to spherical codes, and derive a lower bound on the list size of typical random spherical codes. We also give an upper bound on the list size achievable using nested Construction-A lattices and infinite Construction-A lattices. We then define and study a class of infinite constellations that generalize Construction-A lattices and prove upper and lower bounds for the same. Other goodness properties such as packing goodness and AWGN goodness of infinite constellations are proved along the way. Finally, we consider random lattices sampled from the Haar distribution and show that if a certain conjecture that originates in analytic number theory is true, then the list size grows as a polynomial function of the gap-to-capacity. [ABSTRACT FROM AUTHOR]

The binary quadratic-residue codes and the punctured Reed-Muller codes ${\mathcal {R}}_{2}((m-1)/2, m))$ are two families of binary cyclic codes with parameters $[n, (n+1)/2, d \geq \sqrt {n}]$. These two families of binary cyclic codes are interesting partly due to the fact that their minimum distances have a square-root bound. The objective of this paper is to construct two families of binary cyclic codes of length $2^{m}-1$ and dimension near $2^{m-1}$ with good minimum distances. When $m \geq 3$ is odd, the codes become a family of duadic codes with parameters $[2^{m}-1, 2^{m-1}, d]$ , where $d \geq 2^{(m-1)/2}+1$ if $m \equiv 3 \pmod {4}$ and $d \geq 2^{(m-1)/2}+3$ if $m \equiv 1 \pmod {4}$. The two families of binary cyclic codes contain some optimal binary cyclic codes. [ABSTRACT FROM AUTHOR]

An $(m,n,a,b)$ -tensor code consists of $m\times n$ matrices whose columns satisfy ‘ $a$ ’ parity checks and rows satisfy ‘ $b$ ’ parity checks (i.e., a tensor code is the tensor product of a column code and row code). Tensor codes are useful in distributed storage because a single erasure can be corrected quickly either by reading its row or column. Maximally Recoverable (MR) Tensor Codes, introduced by Gopalan et al., are tensor codes which can correct every erasure pattern that is information theoretically possible to correct. The main questions about MR Tensor Codes are characterizing which erasure patterns are correctable and obtaining explicit constructions over small fields. In this paper, we study the important special case when $a=1$ , i.e., the columns satisfy a single parity check equation. We introduce the notion of higher order MDS codes ($\mathrm {MDS}(\ell)$ codes) which is an interesting generalization of the well-known MDS codes, where $\ell $ captures the order of genericity of points in a low-dimensional space. We then prove that a tensor code with $a=1$ is MR if the row code is an $\mathrm {MDS}(m)$ code. We then show that $\mathrm {MDS}(m)$ codes satisfy some weak duality. Using this characterization and duality, we prove that $(m,n,a=1,b)$ -MR tensor codes require fields of size $q=\Omega _{m,b}(n^{\min \{b,m\}-1})$. Our lower bound also extends to the setting of $a>1$. We also give a deterministic polynomial time algorithm to check if a given erasure pattern is correctable by the MR tensor code (when $a=1$). [ABSTRACT FROM AUTHOR]

This paper studies low-latency streaming codes for the multi-hop network. The source transmits a sequence of messages to a destination through a chain of relays, and requires the destination to reconstruct each message by its deadline. We assume that each communication link is subjected to a certain maximum number of packet erasures. The case of a single relay (a three-node network) was considered in Fong et al. (2020). A coding scheme known as symbol-wise decode and forward was proposed. In the present work, we propose an alternative scheme that is different from Fong et al. (2020) and still achieves the same rate as in Fong et al. (2020) for the one hop case as the field-size goes to infinity. Furthermore, our proposed scheme naturally generalizes to the case of multiple-relay nodes yielding new achievable rates for this setting. The main difference with Fong et al. (2020) is that our proposed scheme exploits the ability of the relay nodes to adapt the transmission based on the erasures on the previous link. Hence, we refer to our scheme as “state-dependent” and contrast it with the scheme in Fong et al. (2020) that is state-independent. Our scheme requires the relay nodes to append a header to the transmitted packets, and we show that the size of the header does not depend on the field-size of the code. We also derive an upper bound on the maximal streaming rate achievable over a network with an arbitrary number of relays. We show that this upper bound matches our achievable rate in the special case when the maximal number of erasures on the first link is greater than or equal to the maximal number of erasures on each of the following links, and the field size goes to infinity. [ABSTRACT FROM AUTHOR]

This paper investigates the secret key-authenticated-capacity region, where information-theoretic authentication is defined by the ability of the decoder to accept and decode messages originating from a valid encoder while rejecting messages from other invalid sources. The model considered here consists of a valid encoder-decoder pairing that can communicate through a channel controlled by an adversary who is also able to eavesdrop on the encoder’s transmissions. Prior to the encoder’s transmission, the adversary decides whether or not to replace the decoder’s observation with an arbitrary one of the adversary’s choosing, with the adversary’s objective being to have the decoder accept and decode their observation to a valid message (different from that of the encoder). To combat the adversary, the encoder and decoder share a secret key. The secret key-authenticated-capacity region is defined as the region of jointly achievable message rate, authentication rate (a to be defined per symbol measure that will generally represent the likelihood that an adversary can fool the decoder), and the key-consumption rate (how many bits of secret key are needed per symbol sent). This is the second of a two-part study, with the parts differing in their measure of the authentication rate. For this second study, the probability of false authentication is considered as a function of the system state, where the system state is defined by the message being transmitted, the value of the secret key, the adversary’s channel observations, and the adversary’s (possibly stochastic) choice for the decoder’s observation. Termed the typical-authentication rate, the authentication measure considered here corresponds to an upper bound on the probability of false authentication for the majority of system states. For this measure, we derive matching inner and outer bounds for the secret key-enabled authenticated capacity region in terms of traditional information-theoretic measures. In doing so, it is shown that the typical-authentication rate and the message rate exhibit a one-to-one trade-off in the capacity region. [ABSTRACT FROM AUTHOR]

In this paper, we establish an interesting duality between two different quantum information-processing tasks, namely, classical source coding with quantum side information, and channel coding over classical-quantum channels. The duality relates the optimal error exponents of these two tasks, generalizing the classical results of Ahlswede and Dueck [IEEE Trans. Inf. Theory, 28(3):430–443, 1982]. We establish duality both at the operational level and at the level of the entropic quantities characterizing these exponents. For the latter, the duality is given by an exact relation, whereas for the former, duality manifests itself in the following sense: an optimal coding strategy for one task can be used to construct an optimal coding strategy for the other task. Along the way, we derive a bound on the error exponent for classical-quantum channel coding with constant composition codes which might be of independent interest. Finally, we consider the task of variable-length classical compression with quantum side information, and a duality relation between this task and classical-quantum channel coding can also be established correspondingly. Furthermore, we study the strong converse of this task, and show that the strong converse property does not hold even in the i.i.d. scenario. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of relaying a single bit of information across a series of binary symmetric channels, and the associated trade-off between the number of hops $m$ , the transmission time $n$ , and the error probability. We introduce a simple, efficient, and deterministic protocol that attains positive information velocity (i.e., a non-vanishing ratio $\frac {m}{n}$ and small error probability) and is significantly simpler than existing protocols that do so. In addition, we characterize the optimal low-noise and high-noise scaling laws of the information velocity, and we adapt our 1-bit protocol to transmit $k$ bits over $m$ hops with $\mathcal {O}(m+k)$ transmission time. [ABSTRACT FROM AUTHOR]

$\mathbb {Z}_{2^{s}}$ -additive codes are subgroups of $\mathbb {Z}^{n}_{2^{s}}$ , and can be seen as a generalization of linear codes over $\mathbb {Z}_{2}$ and $\mathbb {Z}_{4}$. A $\mathbb {Z}_{2^{s}}$ -linear code is a binary code (not necessarily linear) which is the Gray map image of a $\mathbb {Z}_{2^{s}}$ -additive code. We consider $\mathbb {Z}_{2^{s}}$ -additive simplex codes of type $\alpha $ and $\beta $ , which are a generalization over $\mathbb {Z}_{2^{s}}$ of the binary simplex codes. These codes are related to the $\mathbb {Z}_{2^{s}}$ -additive Hadamard codes. In this paper, we use this relationship to find a linear subcode of the corresponding $\mathbb {Z}_{2^{s}}$ -linear codes, called kernel, and a representation of these codes as cosets of this kernel. In particular, this also gives the linearity of these codes. Similarly, $\mathbb {Z}_{2^{s}}$ -additive MacDonald codes are defined for $s>2$ , and equivalent results are obtained. [ABSTRACT FROM AUTHOR]

This paper investigates the secret-key-authenticated-capacity region, where information-theoretic authentication is defined by the ability of the decoder to accept and decode messages originating from a valid encoder while rejecting messages from other invalid sources. The model considered here consists of a valid encoder-decoder pairing that can communicate through a channel controlled by an adversary who is also able to eavesdrop on the encoder’s transmissions. Over multiple rounds of communication, the adversary first decides whether or not to replace the decoder’s observation with an arbitrary one of the adversary’s choosing, with the goal of the adversary being to have the decoder accept and decode their observation as a valid message (different from that of the encoder). To combat the adversary, the encoder and decoder share a secret key. The secret-key-authenticated-capacity region here is then defined as the region of jointly achievable message rate, authentication rate (a to be defined per symbol measure that will generally represent the likelihood that an adversary can fool the decoder), and the key-consumption rate (how many bits of secret key are needed per symbol sent). This is the first of a two-part study, with the parts differing in their measure of the authentication rate. In this first study, the authentication rate is the exponent of blocklength-normalized exponent of the expected probability of false authentication. For this metric, we provide an inner bound which improves on those existing in the literature. This is achieved by adopting and merging different classical techniques in novel ways. Within these classical secret-key-based authentication techniques, one technique derives authentication capability from secure channel coding to send the secret key with the message, and the other technique derives its authentication capability directly from obscuring the source. [ABSTRACT FROM AUTHOR]

In this paper, we determine all dihedral subgroups in $PGL_{2}(\mathbb {F}_{q})$ , where $q=2^{l}$ and $l$ is a positive integer; and construct binary expurgated and extended Goppa codes with dihedral automorphism groups. Moreover, it is easy to obtain binary quasi-cyclic expurgated or extended Goppa codes. [ABSTRACT FROM AUTHOR]

A vertex subset $S$ of a graph $G$ constitutes a 1-perfect code if the one-balls centered at the nodes in $S$ effect a vertex partition of $G$. This paper considers the quad-cube $CQ_{m}$ that is a connected $(m+2)$ -regular spanning subgraph of the hypercube $Q_{4m+2}$ , and shows that $CQ_{m}$ admits a vertex partition into 1-perfect codes iff $m=2^{k}-3$ , where $k\ge 2$. The scheme for that purpose makes use of a procedure by Jha and Slutzki that constructs Hamming codes using a Latin square. The result closely parallels the existence of a 1-perfect code over the dual-cube, which is another derivative of the hypercube. [ABSTRACT FROM AUTHOR]

This article studies the zero-error feedback capacity of causal discrete channels with memory. First, by extending the classical zero-error feedback capacity concept, a new notion of uniform zero-error feedback capacity $C_{0f} $ for such channels is introduced. Using this notion a tight condition for bounded stabilization of unstable noisy linear systems via causal channels is obtained, assuming no channel state information at either end of the channel. Furthermore, the zero-error feedback capacity of a class of additive noise channels is investigated. It is known that for a discrete channel with correlated additive noise, the ordinary capacity with or without feedback is equal $\log q-\mathcal {H}_{ch} $ , where $\mathcal {H}_{ch} $ is the entropy rate of the noise process and $q $ is the input alphabet size. In this paper, for a class of finite-state additive noise channels (FSANCs), it is shown that the zero-error feedback capacity is either zero or $C_{0f} =\log q -h_{ch} $ , where $h_{ch} $ is the topological entropy of the noise process. A condition is given to determine when the zero-error capacity with or without feedback is zero. This, in conjunction with the stabilization result, leads to a “Small-Entropy Theorem”, stating that stabilization over FSANCs can be achieved if the sum of the topological entropies of the linear system and the channel is smaller than $\log q$. [ABSTRACT FROM AUTHOR]

Let $q\geq 2$ be a prime power. In this paper, we study the distribution of the minimum distance (in the Hamming metric) of a random linear code of dimension $k$ in $\mathbb {F}_{q}^{n}$. We provide quantitative estimates showing that the distribution function of the minimum distance is close (superpolynomially in $n$) to the cumulative distribution function of the minimum of $(q^{k}-1)/(q-1)$ independent binomial random variables with parameters $\frac {1}{q}$ and $n$. The latter, in turn, converges to a Gumbel distribution at integer points when $\frac {k}{n}$ converges to a fixed number in (0, 1). Our result confirms in a strong sense that apart from identification of the weights of proportional codewords, the probabilistic dependencies introduced by the linear structure of the random code, produce a negligible effect on the minimum code weight. As a corollary of the main result, we obtain an improvement of the Gilbert–Varshamov bound for $2< q< 49$. [ABSTRACT FROM AUTHOR]

Live communication is ubiquitous, and frequently must contend with reliability issues due to packet loss during transmission. The effect of packet losses can be alleviated by using erasure codes, which aid in recovering lost packets. Streaming codes are a class of codes designed for the live communication setting, which encode a stream of message packets arriving sequentially for transmission over a packet-loss channel. The existing study of streaming codes considers settings where the sizes of the message packets to be transmitted are all fixed. However, message packets occur with unpredictable and variable sizes in many applications, such as videoconferencing. In this paper, we present a generalized model for streaming codes that incorporates message packets of variable sizes. We show that the variability in the sizes of message packets induces a new trade-off between the rate and the decoding delay under lossless transmission. Moreover, the variability in the sizes of message packets impacts the optimal rate of transmission. To address this, we introduce algorithms to compute upper and lower bounds on the optimal rate for any given sequence of sizes of message packets. We then design an explicit streaming code for the proposed model. We empirically evaluate the code construction over a live video trace for several representative parameter settings, and show that the rate of the construction is approximately 90% of an upper bound and 5%–48% higher than naively using the existing streaming codes. [ABSTRACT FROM AUTHOR]

Constrained codes are used to prevent errors from occurring in various data storage and data transmission systems. They can help in increasing the storage density of magnetic storage devices, in managing the lifetime of solid-state storage devices, and in increasing the reliability of data transmission over wires. Over the years, designing practical (complexity-wise) capacity-achieving constrained codes has been an area of research gaining significant interest. We recently designed various constrained codes based on lexicographic indexing. We introduced binary symmetric lexicographically-ordered constrained (S-LOCO) codes, $q$ -ary asymmetric LOCO (QA-LOCO) codes, and a class of two-dimensional LOCO (TD-LOCO) codes. These families of codes achieve capacity with simple encoding and decoding, and they are easy to reconfigure. We demonstrated that these codes can contribute to notable density and lifetime gains in magnetic recording (MR) and Flash systems, and they find application in other systems too. In this paper, we generalize our work on LOCO codes by presenting a systematic method that guides the code designer to build any constrained code based on lexicographic indexing once the finite set of data patterns to forbid is known. In particular, we connect the set of forbidden patterns directly to the cardinality of the LOCO code and most importantly to the rule that uncovers the index associated with a LOCO codeword. By doing that, we reveal the secret arithmetic of patterns, and make the design of such constrained codes significantly easier. We give examples illustrating the method via codes based on lexicographic indexing from the literature. We then design optimal (rate-wise) constrained codes for the new two-dimensional magnetic recording (TDMR) technology. Over a practical TDMR model, we show notable performance gains as a result of solely applying the new codes. Moreover, we show how near-optimal constrained codes for TDMR can be designed and used to further reduce complexity and error propagation. All the newly introduced LOCO codes are designed using the proposed general method, and they inherit all the desirable properties in our previously designed LOCO codes. [ABSTRACT FROM AUTHOR]

Previous works on age of information and erasure channels have dealt with specific models and computed the average age or average peak age for certain settings. In this paper, given a source that produces a letter every $T_{s}$ seconds and an erasure channel that can be used every $T_{c}$ seconds, we ask what is the coding strategy that minimizes the time-average “age of information” that an observer of the channel output incurs. We first analyze the case where the source alphabet and the channel-input alphabet have the same size. We show that a trivial coding strategy is optimal and a closed form expression for the age can be derived. We then analyze the case where the alphabets have different sizes. We use a random coding argument to bound the average age and show that the average age achieved using random codes converges to the optimal average age of linear block codes as the source alphabet becomes large. [ABSTRACT FROM AUTHOR]

Maximum distance separable (MDS) codes are very important in both theory and practice. There is a classical construction of a family of $[{2^{m}+1, 2u-1, 2^{m}-2u+3}]$ MDS codes for $1 \leq u \leq 2^{m-1}$ , which are cyclic, reversible and BCH codes over ${\mathrm {GF}}(2^{m})$. The objective of this paper is to study the quaternary subfield subcodes and quaternary subfield codes of a subfamily of the MDS codes for even $m$. A family of quaternary cyclic codes is obtained. These quaternary codes are distance-optimal in some cases and very good in general. Furthermore, two infinite families of 3-designs from these quaternary codes and their duals are presented. [ABSTRACT FROM AUTHOR]

In this paper we introduce Ouroboros, a new family of Key Exchange protocols based on coding theory. The protocols propose a middle ground between the cryptosystems based on $\mathsf {QC}$ - $\mathsf {MDPC}$ codes, which feature small parameter sizes, but have a security reduction to two problems: the syndrome decoding problem and the indistinguishability of the code, and the HQC protocol, which features bigger parameters but has a security reduction to the syndrome decoding problem only. Ouroboros features a reduction to the syndrome decoding problem with only a small overhead compared to the $\mathsf {QC}$ - $\mathsf {MDPC}$ based cryptosystems. The approach is based on an ideal structure and also works for the rank metric. This yields a simple, secure and efficient approach for key exchange, the Ouroboros family of protocols. For the Hamming metric we obtain the same type of parameters (and almost the same simple decoding) as for $\mathsf {MDPC}$ based cryptosystems, but with a security reduction to decoding random quasi-cyclic codes in the Random Oracle Model. This represents a reduction of up to 38% on the public key size compared to HQC, for the most secure parameters. For the rank metric, we obtain better parameters than for RQC, saving up to 31% on the public key for the most secure set of parameters, using non homogeneous errors in Ouroboros. In this full version, the protocol and decoding algorithm have been slightly improved, additional details are given in the security proof, and the protocol is fully described for the rank metric. [ABSTRACT FROM AUTHOR]

The family of Goppa codes is one of the most interesting subclasses of linear codes. As the McEliece cryptosystem often chooses a random Goppa code as its key, knowledge of the number of inequivalent Goppa codes for fixed parameters may facilitate in the evaluation of the security of such a cryptosystem. In this paper we present a new approach to give an upper bound on the number of inequivalent extended irreducible binary Goppa codes. To be more specific, let $n>3$ be an odd prime number and $q=2^{n}$ ; let $r\geq 3$ be a positive integer satisfying $\gcd (r,n)=1$ and $\gcd \big (r,q(q^{2}-1)\big)=1$. We obtain an upper bound for the number of inequivalent extended irreducible binary Goppa codes of length $q+1$ and degree $r$. [ABSTRACT FROM AUTHOR]

Non-overlapping codes are a set of codewords in $\bigcup _{n \ge 2} \mathbb {Z}_{q}^{n}$ , where $\mathbb {Z}_{q} = \{0,1, {\dots },q-1\}$ , such that the prefix of each codeword is not a suffix of any codeword in the set, including itself; and for variable-length codes, a codeword does not contain any other codeword as a subword. In this paper, we investigate a generic method to generalize binary codes to $q$ -ary ones for $q > 2$ , and analyze this generalization on the two constructions given by Levenshtein (also by Gilbert; Chee, Kiah, Purkayastha, and Wang) and Bilotta, respectively. The generalization on the former construction gives large non-expandable fixed-length non-overlapping codes whose size can be explicitly determined; the generalization on the latter construction is the first attempt to generate $q$ -ary variable-length non-overlapping codes. More importantly, this generic method allows us to utilize the generating function approach to analyze the cardinality of the underlying $q$ -ary non-overlapping codes. The generating function approach not only enables us to derive new results, e.g., recurrence relations on their cardinalities, new combinatorial interpretations for the constructions, and the superior limit of their cardinalities for some special cases, but also greatly simplifies the arguments for these results. Furthermore, we give an exact formula for the number of fixed-length words that do not contain the codewords in a variable-length non-overlapping code as subwords. This thereby solves an open problem by Bilotta and induces a recursive upper bound on the maximum size of variable-length non-overlapping codes. [ABSTRACT FROM AUTHOR]

In this paper we prove several coordinate-ordering-free upper bounds on the insdel distances of linear codes. Our bounds are stronger than some previous known bounds. We apply these upper bounds to AGFC codes from some cyclic codes and one algebraic-geometric code with any rearrangement of coordinate positions. A strong upper bound on the insdel distances of Reed-Muller codes with the special coordinate ordering is also given. [ABSTRACT FROM AUTHOR]

Maximum Distance Separable (MDS) codes with a sparse and balanced generator matrix are appealing in distributed storage systems for balancing and minimizing the computational load. Such codes have been constructed via Reed-Solomon codes over large fields. In this paper, we focus on small fields. We prove that there exists an $[n,k]_{q}$ MDS code that has a sparse and balanced generator matrix for any $q\geq n-1$ provided that $n\leq 2k$ , by designing several algorithms with complexity running in polynomial time in $k$ and $n$. For the case $n>2k$ , we give some constructions for $q=n=p^{s}$ and $k=p^{e}m$ based on sumsets, when $e\leq s-2$ and $m\leq p-1$ , or $e=s-1$ and $m < \frac {p}{2}$. [ABSTRACT FROM AUTHOR]

The strong secrecy transmission problem of the arbitrarily varying wiretap channel (AVWC) with input and state constraints is investigated in this paper. First, a stochastic-encoder code lower bound of the strong secrecy capacity is established by applying the type argument and Csiszár’s almost independent coloring lemma. Then, a superposition stochastic-encoder code lower bound of the secrecy capacity is provided. The superposition stochastic-encoder code lower bound can be larger than the ordinary stochastic-encoder code lower bound. Random code lower and upper bounds of the secrecy capacity of the AVWC with constraints are further provided. Based on these results, we further consider a special case of the model, namely severely less noisy AVWC, and give the stochastic-encoder code and random code capacities. It is proved that the stochastic-encoder code capacity of the AVWC with constraints is either equal to or strictly smaller than the corresponding random code capacity, which is consistent with the property of the ordinary AVC. Finally, some numerical examples are presented to better illustrate our capacity results. Compared to the soft covering lemma that requires the codewords to be generated i.i.d., our method has more relaxed requirements regarding codebooks. It is proved that the good codebooks for secure transmission can be generated by choosing codewords randomly from a given type set, which is critical when considering the AVWC with constraints. [ABSTRACT FROM AUTHOR]

In this paper, we construct four families of linear codes over finite fields from the complements of either the union of subfields or the union of cosets of a subfield, which can produce infinite families of optimal linear codes, including infinite families of (near) Griesmer codes. We also characterize the optimality of these four families of linear codes with an explicit computable criterion using the Griesmer bound and obtain many distance-optimal linear codes. In addition, by a more in-depth discussion on some special cases of these four families of linear codes, we obtain several classes of (distance-)optimal linear codes with few weights and completely determine their weight distributions. It is shown that most of our linear codes are self-orthogonal or minimal which are useful in applications. [ABSTRACT FROM AUTHOR]

Quantum readout provides a general framework for formulating statistical discrimination of quantum channels. Several paths have been taken for such this problem. However, there is much to be done in the avenue of optimizing channel discrimination using classical codes. At least two open questions can be pointed out: how to construct low complexity encoding schemes that are interesting for channel discrimination and, more importantly, how to develop capacity-achieving protocols. This paper aims at presenting a solution to these questions using polar codes. Firstly, we characterize the information rate and reliability parameter of the channels under polar encoding. We also show that the error probability of the scheme proposed decays exponentially with the square root of the code length. Secondly, an analysis of the optimal quantum states to be used as probes is given. [ABSTRACT FROM AUTHOR]

A block code is called a locally recoverable code (LRC for short) with $(r,\delta)$ -locality, if subject to any $\delta -1$ erasure failures, every symbol in the encoding can still be recovered by accessing at most $r$ other symbols. Recently, it was discovered by several authors that a $q$ -ary optimal $(r,\delta)$ -LRC, i.e., an LRC achieving the generalized Singleton-type bound, can have length much bigger than $q+1$. This is quite different from the classical $q$ -ary MDS codes where it is conjectured that the code length is upper bounded by $q+1$ (or $q+2$ for some special cases). In this paper, we further investigate constructions of optimal $(r,\delta)$ -LRCs along the line of using parity-check matrices. Inspired by classical Reed-Solomon codes and the work in Jin (2019), we equip parity-check matrices with the Vandermonde structure. It turns out that a parity-check matrix with the Vandermonde structure that produces an optimal LRC must obey certain disjoint property for subsets of $\mathbb {F}_{q}$. We manage to show the existence of these disjoint subsets. Thus, this yields an optimal $(r,\delta)$ -LRC with code length $\Omega \left({q^{\frac {\delta }{2}\left({1+\frac {1}{\lfloor \frac {d-1}{\delta }\rfloor -1}}\right)}}\right)\vphantom {\bigg)_{j}}$ , where $d$ is the minimum distance. In particular, to our surprise, for $\delta =2$ this disjoint condition is equivalent to a well-studied problem in extremal graph theory. With the help of extremal graph theory, we succeed to improve all of the best known results in Guruswami et al. (2018) for $d\geq 7$. In addition, for $d=6$ , we are able to remove the constraint required in Jin (2019) that $q$ is even. [ABSTRACT FROM AUTHOR]

Linear codes over $\mathbb {Z}_{p^{s}}$ of length $n$ are subgroups of $\mathbb {Z}_{p^{s}}^{n}$. These codes are also called $\mathbb {Z}_{p^{s}}$ -additive codes and can be seen as a generalization of linear codes over $\mathbb {Z}_{2}$ and $\mathbb {Z}_{4}$. A $\mathbb {Z}_{p^{s}}$ -linear code is a code over $\mathbb {Z}_{p}$ , not necessarily linear, which is the generalized Gray map image of a $\mathbb {Z}_{p^{s}}$ -additive code. In 2015, a systematic encoding was found for $\mathbb {Z}_{4}$ -linear codes. Moreover, an alternative permutation decoding method, which is suitable for any binary code (not necessarily linear) with a systematic encoding, was established. In this paper, we generalize these results by presenting a systematic encoding for any $\mathbb {Z}_{p^{s}}$ -linear code with $s\geq 2$ and $p$ prime. We also describe a permutation decoding method for any systematic code over $\mathbb {Z}_{p}$ , not necessarily linear, and show some examples of how to use this systematic encoding in this decoding method. [ABSTRACT FROM AUTHOR]

Combinatorial designs are closely related to linear codes. Recently, some near MDS codes were employed to construct $t$ -designs by Ding and Tang, which settles the question as to whether there exists an infinite family of near MDS codes holding an infinite family of $t$ -designs for $t \geq 2$. This paper is devoted to the construction of infinite families of 3-designs and 2-designs from special equations over finite fields. First, we present an infinite family of almost MDS codes over ${\mathrm{ GF}}(p^{m})$ holding an infinite family of 3-designs. We then provide an infinite family of almost MDS codes over ${\mathrm{ GF}}(p^{m})$ holding an infinite family of 2-designs for any field ${\mathrm{ GF}}(q)$. In particular, some of these almost MDS codes are near MDS. Second, we present an infinite family of near MDS codes over ${\mathrm{ GF}}(2^{m})$ holding an infinite family of 3-designs by considering the number of roots of a special linearized polynomial. Compared to previous constructions of 3-designs or 2-designs from linear codes, the parameters of some of our designs are new and flexible. [ABSTRACT FROM AUTHOR]

In this paper, we present two new families of APN functions. The first family is in bivariate form $\big (x^{3}+xy^{2}+ y^{3}+xy, x^{5}+x^{4}y+y^{5}+xy+x^{2}y^{2} \big)\,\,\vphantom {_{\int _{\int }}}$ over ${\mathbb F}_{2^{m}}^{2}$. It is obtained by adding certain terms of the form $\sum _{i}(a_{i}x^{2^{i}}y^{2^{i}},b_{i}x^{2^{i}}y^{2^{i}})$ to a family of APN functions recently proposed by Gölo&gcaron;lu. The $\vphantom {_{\int _{\int }}}$ second family has the form $L(z)^{2^{m}+1}+vz^{2^{m}+1}$ over ${\mathbb F}_{{2^{3m}}}$ , which generalizes a family of APN functions by Bracken et al. from 2011. By calculating the $\Gamma $ -rank of the constructed APN functions over ${\mathbb F}_{2^{8}}$ and ${\mathbb F}_{2^{9}}$ , we demonstrate that the two families are CCZ-inequivalent to all known families. In addition, the two new families cover two known sporadic APN instances over ${\mathbb F}_{2^{8}}$ and ${\mathbb F}_{2^{9}}$ , which were found by Edel and Pott in 2009 and by Beierle and Leander in 2021, respectively. [ABSTRACT FROM AUTHOR]

One of the important unsolved problems in information theory is the conjecture that the undirected multiple-unicast network information capacity is the same as the routing capacity. This conjecture is verified only for a handful of networks and network classes. Moreover, only two explicit upper bounds on information capacity are known for general undirected networks: the sparsest cut bound and the linear programming bound. In this paper, we present an information-theoretic upper bound, called the partition bound, on the capacity of general undirected multiple-unicast networks. We show that a decision version problem of computing the bound is NP-complete. We present two classes of undirected multiple-unicast networks such that the partition bound is achievable by routing. Thus, the conjecture is proved for these classes of networks. Recently, the conjecture was proved for a new class of networks defined by properties relating to cut-set and source-sink paths. We show the existence of a network outside of this new class of networks such that the partition bound is achievable by routing. [ABSTRACT FROM AUTHOR]