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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]

In an interesting paper, Prof. C. Ding provided three constructions of cyclic codes of length being a product of two primes. Numerical data shows that many codes from these constructions are best cyclic codes of the same length and dimension over the same finite field. However, not much is known about these codes. In this paper, we explain some of the numerical data by developing a general method on cyclic codes of composite length and on estimating the minimum distance. We also provide a general construction of cyclic codes of composite length which are related to Ding’s constructions. Numerical data shows that it produces many best cyclic codes as well. Finally, we point out how these cyclic codes can be used to construct convolutional codes with large free distance. [ABSTRACT FROM AUTHOR]

The paper is devoted to the problem of erasure coding in distributed storage. We consider a model of storage that assumes that nodes are organized into equally sized groups, called racks, that within each group the nodes can communicate freely without taxing the system bandwidth, and that the only information transmission that counts is the one between the racks. This assumption implies that the nodes within each of the racks can collaborate before providing information to the failed node. The main emphasis of the paper is on code construction for this storage model. We present an explicit family of maximum distance separable (MDS) array codes that support recovery of a single failed node from any number of helper racks using the minimum possible amount of inter-rack communication(such codes are said to provide optimal repair). The codes are constructed over finite fields of size comparable to the code length. We also derive a bound on the number of symbols accessed at helper nodes for the purposes of repair, and construct a code family that approaches this bound, while still maintaining the optimal repair property. Finally, we present a construction of scalar Reed-Solomon codes that support optimal repair for the rack-oriented storage model. [ABSTRACT FROM AUTHOR]

Reed-Solomon (RS) codes are widely used in distributed storage systems. In this paper, we study the repair bandwidth and sub-packetization size of RS codes. The repair bandwidth is defined as the amount of transmitted information from surviving nodes to a failed node. The RS code can be viewed as a polynomial over a finite field $GF(q^\ell)$ evaluated at a set of points, where $\ell $ is called the sub-packetization size. Smaller bandwidth reduces the network traffic in distributed storage, and smaller $\ell $ facilitates the implementation of RS codes with lower complexity. Recently, Guruswami and Wootters proposed a repair method for RS codes when the evaluation points are the entire finite field. While the sub-packetization size can be arbitrarily small, the repair bandwidth is higher than the minimum storage regenerating (MSR) bound. Tamo, Ye, and Barg achieved the MSR bound but the sub-packetization size grows faster than the exponential function of the number of the evaluation points. In this paper, we present code constructions and repair schemes that extend these results to accommodate different sizes of the evaluation points. In other words, we design schemes that provide points in between. These schemes provide a flexible tradeoff between the sub-packetization size and the repair bandwidth. In addition, we generalize our schemes to manage multiple failures. [ABSTRACT FROM AUTHOR]

For a function $f$ from $\mathbb {F}_{2}^{n}$ to $\mathbb {F}_{2}^{n}$ , the planarity of $f$ is usually measured by its differential uniformity and differential spectrum. In this paper, we propose the concept of vanishing flats, which supplies a combinatorial viewpoint on the planarity. First, the number of vanishing flats of $f$ can be regarded as a measure of the distance between $f$ and the set of almost perfect nonlinear functions. In some cases, the number of vanishing flats serves as an “intermediate” concept between differential uniformity and differential spectrum, which contains more information than differential uniformity, however less than the differential spectrum. Secondly, the set of vanishing flats forms a combinatorial configuration called partial quadruple system, since it conveys a detailed structural information about $f$. We initiate this study by considering the number of vanishing flats and the partial quadruple systems associated with monomials and Dembowski-Ostrom polynomials. In addition, we present an application of vanishing flats to the partition of a vector space into disjoint equidimensional affine spaces. We conclude the paper with several further questions and challenges. [ABSTRACT FROM AUTHOR]

This paper formulates a distributed computation problem, where a master asks ${\mathsf N}$ distributed workers to compute a linearly separable function. The task function can be expressed as ${\mathsf K}_{\mathrm{ c}}$ linear combinations of ${\mathsf K}$ messages, where each message is a function of one dataset. Our objective is to find the optimal tradeoff between the computation cost (number of uncoded datasets assigned to each worker) and the communication cost (number of symbols the master must download), such that from the answers of any ${\mathsf N}_{\mathrm{ r}}$ out of ${\mathsf N}$ workers the master can recover the task function with high probability, where the coefficients of the ${\mathsf K}_{\mathrm{ c}}$ linear combinations are uniformly i.i.d. over some large enough finite field. The formulated problem can be seen as a generalized version of some existing problems, such as distributed gradient coding and distributed linear transform. In this paper, we consider the specific case where the computation cost is minimum, and propose novel achievability schemes and converse bounds for the optimal communication cost. Achievability and converse bounds coincide for some system parameters; when they do not match, we prove that the achievable distributed computing scheme is optimal under the constraint of a widely used ‘cyclic assignment’ scheme on the datasets. Our results also show that when ${\mathsf K}= {\mathsf N}$ , with the same communication cost as the optimal distributed gradient coding scheme proposed by Tandon et al. from which the master recovers one linear combination of ${\mathsf K}$ messages, our proposed scheme can let the master recover any additional ${\mathsf N}_{\mathrm{ r}}-1$ independent linear combinations of messages with high probability. [ABSTRACT FROM AUTHOR]

Two general constructions of linear codes with functions over finite fields have been extensively studied in the literature. The first one is given by C(ƒ) = {Tr(aƒ(x) + bx)x∈Fqm*: a, b ∈ Fqm}, where q is a prime power, Fqm* = Fqm \ {0}, Tr is the trace function from Fqm to Fq, and ƒ(x) is a function from Fqm to Fqm with ƒ(0) = 0. Almost bent functions, quadratic functions and some monomials on F2m were used in the first construction, and many families of binary linear codes with few weights were obtained in the literature. This paper studies some punctured codes of these binary codes. Several families of binary linear codes with few weights and new parameters are obtained in this paper. Several families of distance-optimal binary linear codes with new parameters are also produced in this paper. [ABSTRACT FROM AUTHOR]

The puncturing and shortening techniques are two important approaches to constructing new linear codes from old ones. In the past 70 years, a lot of progress on the puncturing technique has been made, and many works on punctured linear codes have been done. Many families of linear codes with interesting parameters have been obtained with the puncturing technique. However, little research on the shortening technique has been done and there are only a handful references on shortened linear codes. The first objective of this paper is to prove some general theory for shortened linear codes. The second objective is to study some shortened codes of the Hamming codes, Simplex codes, some Reed-Muller codes, and ovoid codes. Eleven families of optimal shortened codes over finite fields are presented in this paper. As a byproduct, five infinite families of 2-designs are also constructed from some of the shortened codes presented in this paper. [ABSTRACT FROM AUTHOR]

Sidelnikov sequences, a kind of cyclotomic sequences with many desired properties such as low correlation and variable alphabet sizes, can be employed to construct a polyphase sequence family that has many applications in high-speed data communications. Recently, cyclotomic numbers have been used to investigate the linear complexity of Sidelnikov sequences, mainly about binary ones, although the limitation on the orders of the available cyclotomic numbers makes it difficult. This paper continues to study the linear complexity over $\mathbb {F}_{M}$ of $M$ -ary Sidelnikov sequence of period $q-1$ using Hasse derivative, which implies $q=p^{m}$ , $m\geq 1$ and $M|(q-1)$. The $t$ th Hasse derivative formulas are presented in terms of cyclotomic numbers, and some upper bounds on the linear complexity for $M=2$ and 3 are obtained only with some additional restrictions on $q$. Furthermore, concrete illustrations for several families of these sequences, such as $q\equiv 1\pmod {2}$ and $q\equiv 1\pmod {3}$ , show these upper bounds are tight and reachable; especially for $q=2\times 3^{\lambda }+1 (1\leq \lambda \leq 20)$ , the exact linear complexities over $\mathbb {F}_{3}$ of the ternary Sidelnikov sequences are determined; and it turns out that all the linear complexities of the sequences considered are very close to their periods. [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]

This paper generalizes the method proposed by Poulliat et al. for the determination of the optimal Galois field coefficients of a non-binary LDPC parity check constraint based on the binary image of the code. Optimal, or almost-optimal, parity check coefficients are given for check degree varying from 4 to 20 and Galois field varying from GF (64) up to GF (1024). For all given sets of coefficients, no codeword of Hamming weight two exists. A reduced complexity algorithm to compute the binary Hamming weight 3 of a parity check is proposed. When the number of sets of coefficients is too high for an exhaustive search and evaluation, a local greedy search is performed. Explicit tables of coefficients are given. The proposed sets of coefficients can effectively replace the random selection of coefficients often used in NB-LDPC construction. [ABSTRACT FROM AUTHOR]

In addition to their applications in data storage, communications systems, and consumer electronics, linear complementary dual (LCD) codes—a class of linear codes—have been employed in cryptography recently. LCD cyclic codes were referred to as reversible cyclic codes in the literature. The objective of this paper is to construct several families of reversible cyclic codes over finite fields and analyze their parameters. The LCD cyclic codes presented in this paper have very good parameters in general, and contain many optimal codes. A well rounded treatment of reversible cyclic codes is also given in this paper. [ABSTRACT FROM PUBLISHER]

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]

In literatures, there are various constructions of frequency hopping sequence (FHS for short) sets with good Hamming correlations. Some papers employed only multiplicative groups of finite fields to construct FHS sets, while other papers implicitly used only additive groups of finite fields for construction of FHS sets. In this paper, we make use of both multiplicative and additive groups of finite fields simultaneously to present a construction of optimal FHS sets. The construction provides a new family of optimal $\left ({{q}^{{m}}-1,\frac {{q}^{{m}-{t}}-1}{{r}},{\it{ rq}}^{{t}};\frac {{q}^{{m}-{t}}-1}{{r}}+1}\right)$ frequency hopping sequence sets archiving the Peng-Fan bound. Thus, some FHS sets constructed in literatures using either multiplicative groups or additive groups of finite fields are included in our family. In addition, some other FHS sets can be obtained via the well-known recursive construction through one-coincidence sequence set. [ABSTRACT FROM AUTHOR]

In this paper, a class of two-weight and three-weight linear codes over \mathrm GF(p) is constructed, and their application in secret sharing is investigated. Some of the linear codes obtained are optimal in the sense that they meet certain bounds on linear codes. These codes have applications also in authentication codes, association schemes, and strongly regular graphs, in addition to their applications in consumer electronics, communication and data storage systems. [ABSTRACT FROM PUBLISHER]

A code is said to be an $r$ -local locally repairable code (LRC) if each of its coordinates can be repaired by accessing at most r$ other coordinates. When some of the r$ coordinates are also erased, the r$ -local LRC cannot accomplish the local repair, which leads to the concept of (r,\delta)$ -locality. A q is said to have (r, \delta)$ -locality ( \delta \ge 2$ ) if for each coordinate i with support containing $i$ , whose length is at most $r + \delta - 1$ , and whose minimum distance is at least $\delta$ . The $(r, \delta)$ -LRC can tolerate $\delta -1$ erasures in every local code (i.e., punctured subcode), which degenerates to an $r$ -local LRC when $\delta =2$ . A $q$ -ary $(r,\delta)$ LRC is called optimal if it meets the singleton-like bound for $(r,\delta)$ -LRCs. A class of optimal $q$ -ary cyclic $r$ -local LRCs with lengths $n\mid q-1$ were constructed by Tamo, Barg, Goparaju, and Calderbank based on the $q$ -ary Reed-Solomon codes. In this paper, we construct a class of optimal $q$ -ary cyclic $(r,\delta)$ -LRCs ( $\delta \ge 2$ ) with length $n\mid q-1$ , which generalizes the results of Tamo et al. Moreover, we construct a new class of optimal $q$ -ary cyclic $r$ -local LRCs with lengths $n\mid q+1$ and a new class of optimal $q$ -ary cyclic $(r,\delta)$ -LRCs ( $\delta \ge 2$ ) with lengths $n\mid q+1$ . The constructed optimal LRCs with length $n=q+1$ have the best-known length for a given finite field with size $q$ when the minimum distance is larger than 4. [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]

Self-orthogonal codes are an important type of linear codes due to their wide applications in communication and cryptography. The Euclidean (or Hermitian) hull of a linear code is defined to be the intersection of the code and its Euclidean (or Hermitian) dual. It is clear that the hull is self-orthogonal. The main goal of this paper is to obtain self-orthogonal codes by investigating the hulls. Let $\mathcal {C}_{(r,r^{m}-1,\delta,b)}$ be the primitive BCH code over $\mathbb {F}_{r}$ of length $r^{m}-1$ with designed distance $\delta $ , where $\mathbb {F}_{r}$ is the finite field of order $r$. In this paper, we will present Euclidean (or Hermitian) self-orthogonal codes and determine their parameters by investigating the Euclidean (or Hermitian) hulls of some primitive BCH codes. Several sufficient and necessary conditions for primitive BCH codes with large Hermitian hulls are developed by presenting lower and upper bounds on their designed distances. Furthermore, some Hermitian self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated. In addition, we determine the dimensions of the code $\mathcal {C}_{(r,r^{2}-1,\delta,1)}$ and its hull in both Hermitian and Euclidean cases for $2 \le \delta \le r^{2}-1$. We also present two sufficient and necessary conditions on designed distances such that the hull has the largest dimension. [ABSTRACT FROM AUTHOR]

Symbol-pair codes were introduced by Cassuto and Blaum in 2010 to protect pair errors in symbol-pair read channels. Recently Yaakobi, Bruck and Siegel (2016) generalized this notion to b-symbol codes in order to consider consecutive b errors for a prescribed integer b ≥ 2, and they gave constructions and decoding algorithms. Cyclic codes were considered by various authors as candidates for symbol-pair codes and they established minimum distance bounds on (certain) cyclic codes. In this paper we use algebraic curves over finite fields in order to obtain tight lower and upper bounds on b-symbol Hamming weights of arbitrary cyclic codes over Fq. Here b ≥ 2 is an arbitrary prescribed positive integer and Fq is an arbitrary finite field. We also present a stability theorem for an arbitrary cyclic code C of dimension k and length n: the b-symbol Hamming weight enumerator of C is the same as the k-symbol Hamming weight enumerator of C if k ≤ b ≤ n−1. Moreover, we give improved tight lower and upper bounds on b-symbol Hamming weights of some cyclic codes related to irreducible cyclic codes. Throughout the paper the length n is coprime to q. [ABSTRACT FROM AUTHOR]

Pursley and Sarwate established a lower bound on a combined measure of autocorrelation and crosscorrelation for a pair $(f,g)$ of binary sequences (i.e., sequences with terms in {−1, 1}). If $f$ is a nonzero sequence, then its autocorrelation demerit factor, $\text {ADF}(f)$ , is the sum of the squared magnitudes of the aperiodic autocorrelation values over all nonzero shifts for the sequence obtained by normalizing $f$ to have unit Euclidean norm. If $(f,g)$ is a pair of nonzero sequences, then their crosscorrelation demerit factor, $\text {CDF}(f,g)$ , is the sum of the squared magnitudes of the aperiodic crosscorrelation values over all shifts for the sequences obtained by normalizing both $f$ and $g$ to have unit Euclidean norm. Pursley and Sarwate showed that for binary sequences, the sum of $\text {CDF}(f,g)$ and the geometric mean of $\text {ADF}(f)$ and $\text {ADF}{(g)}$ must be at least 1. For randomly selected pairs of long binary sequences, this quantity is typically around 2. In this paper, we show that Pursley and Sarwate’s bound is met for binary sequences precisely when $(f,g)$ is a Golay complementary pair. We also prove a generalization of this result for sequences whose terms are arbitrary complex numbers. We investigate constructions that produce infinite families of Golay complementary pairs, and compute the asymptotic values of autocorrelation and crosscorrelation demerit factors for such families. [ABSTRACT FROM AUTHOR]

Most codes with an algebraic decoding algorithm are derived from Reed-Solomon codes. They are obtained by taking equivalent codes, for example, generalized Reed-Solomon codes, or by using the so-called subfield subcode method, which leads to alternant codes over the underlying prime field, or over some intermediate subfield. The main advantage of these constructions is to preserve both the minimum distance and the decoding algorithm of the underlying Reed-Solomon code. In this paper, we explore in detail the subspace subcodes construction. This kind of codes was already studied in the particular case of cyclic Reed-Solomon codes. We extend this approach to any linear code over the extension of a finite field. We are interested in additive codes who are deeply connected to subfield subcodes. We characterize the duals of subspace subcodes. We introduce the notion of generalized subspace subcodes. We apply our results to generalized Reed-Solomon codes which leads to codes with interesting parameters, especially over a large alphabet. To conclude this paper, we discuss the security of the use of generalized subspace subcodes of Reed-Solomon codes in a cryptographic context. [ABSTRACT FROM AUTHOR]

In this paper, we focus on decoding nonbinary low-density parity-check (LDPC) codes in Galois fields of characteristic two via the proximal alternating direction method of multipliers (proximal-ADMM). By exploiting Flanagan/Constant-Weighting embedding techniques and the decomposition technique based on three-variables parity-check equations, two efficient proximal-ADMM decoders for nonbinary LDPC codes are proposed. We show that both of them are theoretically guaranteed convergent to some stationary point of the decoding model and either of their computational complexities in each proximal-ADMM iteration scales linearly with LDPC code’s length and the size of the considered Galois field. Moreover, the decoder based on the Constant-Weight embedding technique satisfies the favorable property of codeword symmetry. Simulation results demonstrate their effectiveness in comparison with state-of-the-art LDPC decoders. [ABSTRACT FROM AUTHOR]

In an acyclic multicast network, it is well known that a linear network coding solution over GF( q ) exists when q is sufficiently large. In particular, for each prime power q no smaller than the number of receivers, a linear solution over GF( q ) can be efficiently constructed. In this paper, we reveal that a linear solution over a given finite field does not necessarily imply the existence of a linear solution over all larger finite fields. In particular, we prove by construction that: 1) for every \omega \geq 3 , there is a multicast network with source outdegree \omega linearly solvable over GF(7) but not over GF(8), and another multicast network linearly solvable over GF(16) but not over GF(17); 2) there is a multicast network linearly solvable over GF(5) but not over such GF( q ) and over GF( q^m2 ) is not necessarily linearly solvable over GF( q^{m1 + m2} ); and 4) there exists a class of multicast networks with a set T of receivers such that the minimum field size q_{\mathrm{ min}} for a linear solution over GF( q_{\mathrm{ min}} ) is lower bounded by \Theta (\sqrt T) , but not every larger field than GF( q\mathrm{ min} ) suffices to yield a linear solution. The insight brought from this paper is that not only the field size but also the order of subgroups in the multiplicative group of a finite field affects the linear solvability of a multicast network. [ABSTRACT FROM AUTHOR]

Vector linear solvability of non-multicast networks depends upon both the characteristic of the finite field and the dimension of the vector linear network code. In the literature, the dependency on the characteristic of the finite field and the dependency on the dimension have been studied separately. In this paper, we show the interdependency between the characteristic of the finite field and the dimension of the vector linear network code that achieves a vector linear network coding (VLNC) solution in non-multicast networks. For any given network $\mathcal {N}$ , we define $P(\mathcal {N},d)$ as the set of all characteristics of finite fields over which the network $\mathcal {N}$ has a $d$ -dimensional VLNC solution. To the best of our knowledge, for any network $\mathcal {N}$ shown in the literature, if $P(\mathcal {N},1)$ is non-empty, then $P(\mathcal {N},1) = P(\mathcal {N},d)$ for any positive integer $d$. We show that, for any two non-empty sets of primes $P_{1}$ and $P_{2}$ , there exists a network $\mathcal {N}$ such that $P(\mathcal {N},1) = P_{1}$ , but $P(\mathcal {N},2) = \{P_{1},P_{2} \}$. We also show that there are networks exhibiting a similar advantage (the existence of a VLNC solution over a larger set of characteristics) if the dimension is increased from 2 to 3. However, such behaviour is not universal, as there exist networks which admit a VLNC solution over a smaller set of characteristics of finite fields when the dimension is increased. Using the networks constructed in this paper, we further demonstrate that: (i) a network having an $m_{1}$ -dimensional VLNC solution over a finite field of some characteristic and an $m_{2}$ -dimensional VLNC solution over a finite field of some other characteristic may not have an $(m_{1} + m_{2})$ -dimensional VLNC solution over any finite field; (ii) there exist a class of networks for which scalar linear network coding (SLNC) over non-commutative rings has some advantage over SLNC over finite fields: the least sized non-commutative ring over which each network in the class has an SLNC solution is significantly lesser in size than the least sized finite field over which it has an SLNC solution. [ABSTRACT FROM AUTHOR]

Cyclic codes have been widely used in many applications of communication systems and data storage systems. This paper proposes a new procedure for decoding cyclic codes up to actual minimum distance. The decoding procedure consists of two steps: 1) computation of known syndromes and 2) computation of error positions and error values simultaneously. To do so, a matrix whose all entries are syndromes is called syndrome matrix. A matrix whose entries are either syndromes or the elements of a finite field is said to be partial syndrome matrix. In this paper, two novel methods are presented to determine error positions and error values simultaneously and directly. The first method uses a new partial syndrome matrix along with Gaussian elimination. The partial syndrome matrices for binary (respectively, ternary) cyclic codes of lengths from 69 to 99 (respectively, 16 to 37) are tabulated. For some cyclic codes, the partial syndrome matrices contain unknown syndromes; the second method constructs a matrix from a system of equations, which is generated by the determinants of different partial syndrome matrices and makes use of Gaussian elimination to determine its row rank. Many more cyclic codes beyond the Bose–Chaudhuri–Hocquenghem bound can be decoded with these methods. [ABSTRACT FROM AUTHOR]

We study the communication efficient secret sharing (CESS) problem. A classical threshold secret sharing scheme encodes a secret into $n$ shares given to n$ parties, such that any set of at least t$ , t n$ . However, Staircase codes may require dividing the secret and the shares into many symbols. We also describe how Staircase codes can be used to construct threshold changeable secret sharing with minimum storage cost, i.e., minimum share size. [ABSTRACT FROM AUTHOR]

In practical applications, such as direct spread code division multiple access communications, space-time codes and compressed sensing, and codebooks with small inner-product correlation are required. It is extremely difficult to construct codebooks achieving the Levenshtein bound. In this paper, two new constructions of infinitely many codebooks with multiplicative characters of finite fields are presented. These constructions produce complex codebooks asymptotically achieving the Levenshtein bound and codebooks asymptotically achieving the Welch bound. The codebooks presented in this paper have new parameters. [ABSTRACT FROM PUBLISHER]

Pairs of binary sequences formed using linear combinations of multiplicative characters of finite fields are exhibited that, when compared with a random sequence pairs, simultaneously achieve significantly lower mean square autocorrelation values (for each sequence in the pair) and significantly lower mean square crosscorrelation values. If we define crosscorrelation merit factor analogously to the usual merit factor for autocorrelation, and if we define demerit factor as the reciprocal of merit factor, then randomly selected binary sequence pairs are known to have an average crosscorrelation demerit factor of 1. Our constructions provide sequence pairs with a crosscorrelation demerit factor significantly less than 1, and at the same time, the autocorrelation demerit factors of the individual sequences can also be made significantly less than 1 (which also indicates better than average performance). The sequence pairs studied here provide combinations of autocorrelation and crosscorrelation performance that are not achievable using sequences formed from single characters, such as maximal linear recursive sequences (m-sequences) and Legendre sequences. In this paper, exact asymptotic formulae are proved for the autocorrelation and crosscorrelation merit factors of sequence pairs formed using linear combinations of multiplicative characters. Data is presented that shows that the asymptotic behavior is closely approximated by sequences of modest length. [ABSTRACT FROM PUBLISHER]

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]

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]

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]

Sum-rank metric codes have recently attracted the attention of many researchers, due to their relevance in several applications. Mathematically, the sum-rank metric is a natural generalization of both the Hamming metric and the rank metric. In this paper, we provide an Anticode Bound for the sum-rank metric, which extends the corresponding Hamming and rank-metric Anticode bounds. We classify then optimal anticodes, i.e., codes attaining the sum-rank metric Anticode Bound. We use these optimal anticodes to define generalized sum-rank weights and we study their main properties. In particular, we prove that the generalized weights of an MSRD code are determined by its parameters. As an application, in the Appendix we explain how generalized weights measure information leakage in multishot network coding. [ABSTRACT FROM AUTHOR]

Highly nonlinear functions (perfect nonlinear, maximum nonlinear, etc.) between finite fields have been studied in numerous papers. These concepts have been generalized first to finite abelian groups and then to finite nonabelian groups. In 2011, Poinsot and Pott introduced a new class of highly nonlinear functions between finite nonabelian groups, which are closely related to maximum nonlinear functions. They found such a function by a computer search, and proposed to find more such functions by theoretical constructions. Since then there is no progress in this topic. In this paper we continue the research in the direction proposed by Poinsot and Pott. Our purpose is to study the properties and constructions of this new class of highly nonlinear functions, called the absolute maximum nonlinear functions in this paper. In particular, for arbitrary finite (abelian or nonabelian) groups $K$ and $N$ with an absolute maximum nonlinear function $f: K \to N$ , we will prove that $N$ has to be abelian, and $f$ cannot be perfect nonlinear if $K$ is not abelian. Then we investigate the numerical constraints on the groups that have absolute maximum nonlinear functions. These constraints will eliminate the existence of absolute maximum nonlinear functions for many finite nonabelian groups. More importantly, using these constraints we will develop a method to construct absolute maximum nonlinear functions for some groups. In particular, we will determine all absolute maximum nonlinear functions on $A_4$ (the alternating group of degree 4). Furthermore, a general method to construct new absolute maximum nonlinear functions from the old ones will be given. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of binary nonlinear feedback shift register sequences generated by cascaded feedback registers, one is an LFSR and the other one generates a de Bruijn sequence. The cycle structure (in particular, the initial state of each cycle) is determined by solving a system of linear equations. As an application, we can generate de Bruijn sequences of large period algorithmically. [ABSTRACT FROM AUTHOR]

Let $\mathbb {F}_{q}$ be the finite field of $q$ elements. In this paper we obtain bounds on the following counting problem: given a polynomial $f(x)\in \mathbb {F}_{q} [x]$ of degree $k+m$ and a non-negative integer $r$ , count the number of polynomials $g(x)\in \mathbb {F}_{q} [x]$ of degree at most $k-1$ such that $f(x)+g(x)$ has exactly $r$ roots in $\mathbb {F}_{q}$. Previously, explicit formulas were known only for the cases $m=0, 1, 2$. As an application, we obtain an asymptotic formula on the list size of the standard Reed-Solomon code $[q, k, q-k+1]_{q}$. [ABSTRACT FROM AUTHOR]

In this paper, we consider a new normalization of matrices obtained by choosing distinct codewords at random from linear codes over finite fields and find that under some natural algebraic conditions of the codes their empirical spectral distribution converges to Wigner’s semicircle law as the length of the codes goes to infinity. One such condition is that the dual distance of the codes is at least five. This is analogous to previous work on the empirical spectral distribution of similar matrices obtained in this fashion that converges to the Marchenko–Pastur law. [ABSTRACT FROM AUTHOR]

For any prime power $q$ , Mori and Tanaka introduced a family of $q$ -ary polar codes based on the $q\,\,\times \,\,q$ Reed–Solomon polarization kernels. For transmission over a $q$ -ary erasure channel, they also derived a closed-form recursion for the erasure probability of each effective channel. In this paper, we use that expression to analyze the finite-length scaling of these codes on the $q$ -ary erasure channel with erasure probability $\epsilon \in (0,1)$. Our primary result is that for any $\gamma >0$ and $\delta >0$ , there is a $q_{0}$ , such that for all $q\geq q_{0}$ , the fraction of effective channels with erasure rate at most $N^{-\gamma }$ is at least $1-\epsilon -O(N^{-1/2+\delta })$ , where $N=q^{n}$ is the blocklength. Since this fraction cannot be larger than $1-\epsilon -O(N^{-1/2})$ , this establishes near-optimal finite-length scaling for this family of codes. Our approach can be seen as an extension of a similar analysis for binary polar codes by Hassani, Alishahi, and Urbanke. A similar analysis is also considered for $q$ -ary polar codes with $m \times m$ polarizing matrices. This separates the effect of the alphabet size from the effect of the matrix size. If the polarizing matrix at each stage is drawn independently and uniformly from the set of invertible $m \times m$ matrices, then the linear operator associated with the Lyapunov function analysis can be written in the closed form. To prove near-optimal scaling for polar codes with fixed $q$ as $m$ increases, however, two technical obstacles remain. Thus, we conclude by stating two concrete mathematical conjectures that, if proven, would imply near-optimal scaling for fixed $q$. [ABSTRACT FROM AUTHOR]

Due to wide applications of binary sequences with a low correlation to communications, various constructions of such sequences have been proposed in the literature. Many efforts have been made to construct good binary sequences with various lengths. However, most of the known constructions make use of the multiplicative cyclic group structure of finite field $\mathbb {F}_{2^{n}}$ for a positive integer $n$. It is often overlooked in this community that all $2^{n}+1$ rational places (including “the place at infinity”) of the rational function field over $\mathbb {F}_{2^{n}}$ form a cyclic structure under an automorphism of order $2^{n}+1$. In this paper, we make use of this cyclic structure to provide an explicit construction of binary sequences with a low correlation of length $2^{n}+1$ via cyclotomic function fields over $\mathbb {F}_{2^{n}}$. Each family of our sequences has size $2^{n}-1$ and its correlation is upper bounded by $\lfloor 2^{(n+2)/2}\rfloor $. To the best of our knowledge, this is the first construction of binary sequences with a low correlation of length $2^{n}+1$. Moreover, our sequences can be constructed explicitly and have competitive parameters. In particular, compared with the Gold sequences of length $2^{n}-1$ for even $n$ , our sequences have a smaller correlation and a larger length although the family size of our sequences is slightly smaller. [ABSTRACT FROM AUTHOR]

Minimal linear codes have significant applications in secret sharing schemes and secure two-party computation. When they are defined over finite fields, those codes have been intensively studied, especially in recent years, but they have been firstly partially characterized by Ashikhmin and Barg since 1998. Next, they were completely characterized in 2018 by Ding, Heng, and Zhou in terms of the minimum and maximum nonzero weights in the corresponding codes. Since then, many construction methods for minimal linear codes over finite fields throughout algebraic and geometric approaches have been proposed in the literature. In particular, the algebraic approach gives rise to minimal codes from (cryptographic) functions. Linear codes over finite fields have been expanded into the collection of acceptable alphabets for codes and study codes over finite commutative rings. A natural way to extend the known results available in the literature is to consider minimal linear codes over commutative rings with unity. In extending coding theory to codes over rings, several essential principles must be considered. Particularly extending the minimality property from finite fields to rings and creating such codes is not simple. Such an extension offers more flexibility in the construction of minimal codes. The present article investigates one-dimensional minimal linear codes over the rings $\mathbb {Z}_{p^{n}}$ (where $p$ is a prime) and $\mathbb {Z}_{p^{m}q^{n}}$ (where $p < q$ are distinct primes and $m\leq n$). Our ultimate objective is to characterize such codes’ minimality and design minimal linear codes over the considered rings. Given our objective, we first introduced the notion of minimal codes over (commutative) rings and succeeded in deriving simple characterization of one-dimensional minimal linear codes over the underlying rings mentioned above. Our new algebraic approach allows designing new minimal linear codes. Almost minimal codes over rings are also presented. To the best of our knowledge, the present paper offers a wide variety of minimal codes over (commutative) rings for the first time. Novel perspectives and developments in this direction are expected in the future. [ABSTRACT FROM AUTHOR]

Just as rank-metric or Gabidulin codes may be used to construct rate–diversity tradeoff optimal space–time codes, a recently introduced generalization for the sum-rank metric—linearized Reed–Solomon codes—accomplishes the same in the case of multiple fading blocks. In this paper, we provide the first explicit construction of minimal delay rate–diversity optimal multiblock space–time codes as an application of linearized Reed–Solomon codes. We also provide sequential decoders for these codes and, more generally, space–time codes constructed from finite field codes. Simulation results show that the proposed codes can outperform full diversity codes based on cyclic division algebras at low SNRs as well as utilize significantly smaller constellations. [ABSTRACT FROM AUTHOR]

In this paper, list decoding of crisscross errors in arrays over finite fields is considered. For this purpose, the so-called cover metric is used, where the cover of a matrix is a set of rows and columns which contains all non-zero elements of the matrix. A Johnson-like upper bound on the maximum list size in the cover metric is derived, showing that the list of codewords has polynomial size up to a certain radius. Furthermore, a simple list decoding algorithm for a known optimal code construction is presented, which decodes errors in the cover metric up to our upper bound. These results reveal significant differences between the cover metric and the rank metric and show that the cover metric is more suitable for correcting crisscross errors. [ABSTRACT FROM PUBLISHER]

Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. Minimal linear codes have been studied since decades but their tight connection with cutting blocking sets of finite projective spaces was unfolded only in the past few years, and it has not been fully exploited yet. In this paper we apply finite geometric and probabilistic arguments to contribute to the field of minimal codes. We prove an upper bound on the minimal length of minimal codes of dimension $k$ over the $q$ -element Galois field which is linear in both $q$ and $k$ , hence improve the previous superlinear bounds. This result determines the minimal length up to a small constant factor. We also improve the lower and upper bounds on the size of so called higgledy-piggledy line sets in projective spaces and apply these results to present improved bounds on the size of covering codes and saturating sets in projective spaces as well. [ABSTRACT FROM AUTHOR]

Let $n (>3)$ be a prime number and ${\mathbb {F}}_{2^{n}}$ a finite field of $2^{n}$ elements. Let $L ={\mathbb {F}}_{2^{n}}\cup \{\infty \}$ be the support set and $g(x)$ an irreducible polynomial of degree 6 over ${\mathbb {F}}_{2^{n}}$. In this paper, we obtain an upper bound on the number of extended irreducible binary Goppa codes $\Gamma (L, g)$ of degree 6 and length $2^{n}+1$. [ABSTRACT FROM AUTHOR]

The $q^{t} \times (q+1)t$ ordered orthogonal arrays (OOAs) of strength $t$ over the alphabet $ \mathbb {F}_{q}$ were constructed using linear feedback shift register sequences (LFSRs) defined by primitive polynomials in $ \mathbb {F}_{q}[x]$. In this paper, we extend this result to all polynomials in $\mathbb {F}_{q}[x]$ which satisfy some fairly simple restrictions, i.e., the restrictions that are automatically satisfied by primitive polynomials. While these restrictions sometimes reduce the number of columns produced from $(q+1)t$ to a smaller multiple of $t$ , in many cases, we still obtain the maximum number of columns in the constructed OOA when using non-primitive polynomials. For $2 \le q \le 9$ and small $t$ , we generate OOAs in this manner for all permissible polynomials of degree $t$ in $ \mathbb {F}_{q}[x]$ and compare the results to the ones produced in , , and showing how close the arrays are to being “full” orthogonal arrays. Unusually for the finite fields, our arrays based on the non-primitive irreducible and even reducible polynomials are closer to the orthogonal arrays than those built from the primitive polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we construct some class of explicit subcovers of the curve $\mathcal {X}_{n,r}$ defined over $\mathbb {F}_{q^{n}}$ by affine equation $y^{q^{n-1}}+\cdots +y^{q}+y=x^{q^{n-r}+1}-x^{q^{n}+q^{n-r}}$. These subcovers are defined over $\mathbb {F}_{q^{n}}$ by affine equation $g_{s}(y)=x^{q^{n}+q^{n-r}}-x^{q^{n-r}+1}$ , where $g_{s}(y)$ is a $q$ -polynomial of degree $q^{s}$. The Weierstrass semigroup $H(P_\infty)$ , where $P_\infty $ is the only point at infinity on such subcovers, is determined for $1 \leq s \leq 2r-n+1$ , and the corresponding one-point AG codes are investigated. Codes establishing new records on the parameters with respect to the previously known ones are discovered, and 108 improvements on MinT tables are obtained. [ABSTRACT FROM AUTHOR]

The study of regenerating codes has advanced tremendously in recent years. However, most known constructions require large field size and, hence, may be hard to implement in practice. By restructuring a code construction by Rashmi et al., we obtain two explicit families regenerating codes. These codes approach the cut-set bound as the reconstruction degree increases and may be realized over any given finite field if the file size is large enough. Essentially, these codes constitute a constructive proof that the cut-set bound does not imply a field size restriction, unlike some known bounds for ordinary linear codes. The first construction attains the cut-set bound at the MBR point asymptotically for all parameters, whereas the second one attains the cut-set bound at the MSR point asymptotically for low-rate parameters. Even though these codes require a large file size, this restriction is trivially satisfied in most conceivable distributed storage scenarios, that are the prominent motivation for regenerating codes. [ABSTRACT FROM AUTHOR]

In 1964, Massey introduced a class of codes with complementary duals which are called linear complimentary dual (LCD) codes. He showed that LCD codes have applications in communication system, side-channel attack and so on. LCD codes have been extensively studied in literature. On the other hand, MDS codes form an optimal family of classical codes which have wide applications in both theory and practice. The main purpose of this paper is to give an explicit construction of several classes of LCD MDS codes, using tools from algebraic function fields. We exemplify this construction and obtain several classes of explicit LCD MDS codes for the odd characteristic case. [ABSTRACT FROM AUTHOR]

It was shown by Massey that linear complementary dual (LCD) codes are asymptotically good. In 2004, Sendrier proved that LCD codes meet the asymptotic Gilbert–Varshamov (GV) bound. Until now, the GV bound still remains to be the best asymptotical lower bound for LCD codes. In this paper, we show that an algebraic geometry code over a finite field of even characteristic is equivalent to an LCD code and consequently there exists a family of LCD codes that are equivalent to algebraic geometry codes and exceed the asymptotical GV bound. [ABSTRACT FROM AUTHOR]

In this paper, by the Hasse-Weil bound, we determine the necessary and sufficient condition on coefficients $a_{1},a_{2},a_{3}\in {\mathbb F} _{2^{n}}$ with $n=2m$ such that $f(x) = {x}^{3\cdot 2^{m}} + a_{1}x^{2^{m+1}+1} + a_{2} x^{2^{m}+2} + a_{3}x^{3}$ is an APN function over ${\mathbb F}_{2^{n}}$. Our work together with the follow-up work by Chase and Lisoněk indicates that all such APN quadrinomials $f(x)$ are affine equivalent to two instances of Gold functions, which resolves the first half of an open problem by Carlet at the International Workshop on the Arithmetic of Finite Fields, 83-107, 2014. [ABSTRACT FROM AUTHOR]

A linear error correcting code is a subspace of a finite-dimensional space over a finite field with a fixed coordinate system. Such a code is said to be locally recoverable with locality r if, for every coordinate, its value at a codeword can be deduced from the value of (certain) r other coordinates of the codeword. These codes have found many recent applications, e.g., to distributed cloud storage. We will discuss the problem of constructing good locally recoverable codes and present some constructions using algebraic surfaces that improve previous constructions and sometimes provide codes that are optimal in a precise sense. The main conceptual contribution of this paper is to consider surfaces fibered over a curve in such a way that each recovery set is constructed from points in a single fiber. This allows us to use the geometry of the fiber to guarantee the local recoverability and use the global geometry of the surface to get a hold on the standard parameters of our codes. We look in detail at situations where the fibers are rational or elliptic curves and provide many examples applying our methods. [ABSTRACT FROM AUTHOR]