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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]

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]

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]

Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of BCH codes $\mathcal {C}_{(q,q+1,4,1)}$ and their dual codes with $q=2^{m}$ and established that the codewords of the minimum (or the second minimum) weight in these codes support 4-designs or 3-designs. Motivated by this, we further investigate the codewords of the next adjacent weight in such codes and discover more infinite classes of $t$ -designs with $t=3,4$. In particular, we prove that codewords of weight 7 in $\mathcal {C}_{(q,q+1,4,1)}$ support 4-designs for odd $m \geqslant 5$ and they support 3-designs for even $m \geqslant 4$ , which provide infinite classes of simple $t$ -designs with new parameters. Another significant class of $t$ -designs we produce in this paper has complementary designs with parameters 4- $(2^{2s+1}+ 1,5,5)$ ; these designs have the smallest index among all the known simple 4- $(q+1,5,\lambda)$ designs derived from codes for prime powers $q$ ; and they are further proved to be isomorphic to the 4-designs admitting the projective general linear group PGL $(2,2^{2s+1})$ as the automorphism group constructed by Alltop in 1969. [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]

Previous algorithms for decoding AG codes up to the designed distance all assumed existence of an extra rational place on the base algebraic curve. The place is used to solve the decoding problem by linear algebra over the base field of the curve. The rationality of the place is essential, and therefore AG codes supported by all rational places on the curve are excluded from the domain of applicability of the decoding algorithms. This paper presents a decoding algorithm for those AG codes using an extra place of higher degree. Hence finally all AG codes, as Goppa defined 40 years ago, are equipped with a fast decoding algorithm. [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]

Sequences and their correlation properties have been extensively studied due to their broad applications. In this paper, we develop a connection between sequences and well-studied combinatorial objects, circular Florentine arrays. This connection allows us to derive two types of sequences with good correlation properties. The first type consists of sequences having optimal correlation with respect to the Sarwate bound. Our constructions are based on perfect polyphase sequences. The number of perfect sequences with optimal correlation depends on the existence of circular Florentine arrays, which improves the previous known results. The second type is about multiple ZCZ sequence sets with low inter-set cross-correlation. Each generated ZCZ sequence set is optimal with respect to the Tang-Fan-Matsufuji bound, and each sequence in each set is perfect. In addition, any two sequences from distinct ZCZ sequence sets possess optimal inter-set cross-correlation with respect to the Sarwate bound. Compared with the previous results, the number of ZCZ sequence sets with optimal inter-set cross-correlation property is improved, because of the existence of circular Florentine arrays. [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]

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]

Both linear complementary dual (LCD) codes and maximum distance separable (MDS) codes have good algebraic structures, and they have interesting practical applications such as communication systems, data storage, quantum codes, and so on. So far, most of LCD MDS codes have been constructed by employing generalized Reed-Solomon codes. In this paper we construct some classes of new Euclidean LCD MDS codes and Hermitian LCD MDS codes which are not monomially equivalent to Reed-Solomon codes, called LCD MDS codes of non-Reed-Solomon type. Our method is based on the constructions of Beelen et al. (2017) and Roth and Lempel (1989). To the best of our knowledge, this is the first paper on the construction of LCD MDS codes of non-Reed-Solomon type; any LCD MDS code of non-Reed-Solomon type constructed by our method is not monomially equivalent to any LCD code constructed by the method of Carlet et al. (2018). [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]

$\mathbb {Z}_{2^{s}}$ -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]

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]

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]

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]

This paper investigates the decoding of a remarkable set of lattices: We treat in a unified framework the Leech lattice in dimension 24, the Nebe lattice in dimension 72, and the Barnes-Wall lattices. A new interesting lattice, named $L_{3\cdot 24}$ , is constructed as a simple application of the single parity check on the Leech lattice. The common aspect of these lattices is that they can be obtained via a single parity check or via the $k$ -ing construction. We exploit these constructions to introduce a new efficient paradigm for decoding. This leads to efficient list decoders and quasi-optimal decoders on the Gaussian channel. Both theoretical and practical performance (point error probability and complexity) of the new decoders are provided. [ABSTRACT FROM AUTHOR]

In this article, we present a new construction of evaluation codes in the Hamming metric, which we call twisted Reed–Solomon codes. Whereas Reed–Solomon (RS) codes are MDS codes, this need not be the case for twisted RS codes. Nonetheless, we show that our construction yields several families of MDS codes. Further, for a large subclass of (MDS) twisted RS codes, we show that the new codes are not generalized RS codes. To achieve this, we use properties of Schur squares of codes as well as an explicit description of the dual of a large subclass of our codes. We conclude the paper with a description of a decoder, that performs very well in practice as shown by extensive simulation results. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study quaternary linear codes and their binary subfield codes. First we obtain a general explicit relationship between quaternary linear codes and their binary subfield codes in terms of generator matrices and defining sets. Second, we construct quaternary linear codes via simplicial complexes and determine the weight distributions of these codes. Third, the weight distributions of the binary subfield codes of these quaternary codes are also computed by employing the general characterization. Furthermore, we present two infinite families of optimal linear codes with respect to the Griesmer Bound, and a class of binary almost optimal codes with respect to the Sphere Packing Bound. We also need to emphasize that we obtain at least 9 new quaternary linear codes. [ABSTRACT FROM AUTHOR]

Local pseudorandom generators are a class of fundamental cryptographic primitives having very broad applications in theoretical cryptography. Following Couteau et al.’s work at ASIACRYPT 2018, this paper further studies the concrete security of one important class of local pseudorandom generators, i.e., Goldreich’s pseudorandom generators. Our first attack is of the guess-and-determine type. Our result significantly improves the state-of-the-art algorithm proposed by Couteau et al., in terms of both asymptotic and concrete complexity, and breaks all the challenge parameters they proposed. For instance, for a parameter set suggested for 128 bits of security, we could solve the instance faster by a factor of about 277, thereby destroying the claimed security completely. Our second attack further exploits the extremely sparse structure of the predicate $P_{5}$ and combines ideas from iterative decoding. This novel attack, named guess-and-decode, substantially improves the guess-and-determine approaches for cryptographic-relevant parameters. All the challenge parameter sets proposed in Couteau et al.’s work in ASIACRYPT 2018 aiming for 80-bit (128-bit) security levels can be solved in about 258 (278) operations. We suggest new parameters for achieving 80-bit (128-bit) security with respect to our attacks. We also extend the attacks to other promising predicates and investigate their resistance. [ABSTRACT FROM AUTHOR]

As a special subclass of cyclic codes, BCH codes have wide applications in communication and storage systems. A BCH code of length $n$ over $\mathbb {F}_{q}$ is always relative to an $n$ -th primitive root of unity $\beta $ in an extension field of $\mathbb {F}_{q}$ , and is called a dually-BCH code if its dual is also a BCH code relative to the same $\beta $. The question as to whether a BCH code is a dually-BCH code is in general very hard to answer. In this paper, an answer to this question for primitive narrow-sense BCH codes and projective narrow-sense ternary BCH codes is given. Sufficient and necessary conditions in terms of the designed distances $\delta $ will be presented to ensure that these BCH codes are dually-BCH codes. In addition, the parameters of the primitive narrow-sense BCH codes and their dual codes are investigated. Some lower bounds on minimum distances of the dual codes of primitive and projective narrow-sense BCH codes are developed. Especially for binary primitive narrow-sense BCH codes, the new bounds on the minimum distances of the dual codes improve the classical Sidel’nikov bound, and are also better than the Carlitz and Uchiyama bound for large designed distances $\delta $. The question as to what subclasses of cyclic codes are BCH codes is also answered to some extent. As a byproduct, the parameters of some subclasses of cyclic codes are also investigated. [ABSTRACT FROM AUTHOR]

This paper considers the transmission of an infinite sequence of messages (a streaming source) over a packet erasure channel, where every source message must be recovered perfectly at the destination subject to a fixed decoding delay. While the capacity of a channel that introduces only bursts of erasures has been known for more than fifteen years, only recently, the capacity of a channel with either one burst of erasures or multiple arbitrary erasures in any fixed-sized sliding window has been established. However, explicit codes shown to achieve this capacity for any admissible set of parameters require a large field size that scales exponentially with the delay. Recently, it has been shown that there exist codes that achieve the capacity with field size which scales quadratically with the delay. However, explicit constructions were shown only to specific combinations of parameters. This work describes an explicit rate-optimal construction for all admissible channel and delay parameters over a field size that scales quadratically with the delay. [ABSTRACT FROM AUTHOR]

This paper proposes the Guruswami-Sudan (GS) list decoding algorithm for one-point elliptic codes, in which the interpolation is realized by the module basis reduction (BR). Elliptic codes are a kind of algebraic-geometric (AG) codes with a genus of one. Over the same finite field, they have a greater codeword length than Reed-Solomon (RS) codes, capable of correcting more errors. The GS decoding consists of interpolation and root-finding, while the former that determines the interpolation polynomial $\mathcal {Q}(\text {x}, \text {y}, \text {z})$ dominates the decoding complexity. By defining the Lagrange interpolation function over an elliptic function field, a basis of the interpolation module can be constructed. The desired Gröbner basis that contains $\mathcal {Q}(\text {x}, \text {y}, \text {z})$ can be determined by reducing the constructed basis. This is namely the BR interpolation and it requires less finite field arithmetic operations than the conventional Kötter’s interpolation, facilitating the GS decoding. Re-encoding transform (ReT) is further introduced to facilitate the BR interpolation. This work also shows that both the BR interpolation and its ReT variant will have a lower complexity as the code rate ${k}/{n}$ increases, where n and ${k}$ are the length and dimension of the code, respectively. Our numerical results demonstrate the complexity advantage of the BR interpolation over Kötter’s interpolation, and the performance advantage of elliptic codes over RS codes. [ABSTRACT FROM AUTHOR]

A Goppa code over $\Bbb F_{q^{m}}$ is a well-known subclass of algebraic error-correcting code. If $m=1$ , then it is a generalized Reed-Solomon(GRS) code and its dual code is called a GRS code via a Goppa code. In this paper, we give a necessary and sufficient condition that the dual codes of GRS codes via (expurgated) Goppa codes are also GRS codes via Goppa codes. Under the above condition, we show that the hulls of GRS codes via Goppa codes are still GRS codes via Goppa codes. As an application, we characterize LCD GRS codes and self-dual GRS codes under the above condition. Some numerical examples are also presented to illustrate our main results. Moreover, we also apply our result to entanglement-assisted quantum error correcting codes (EAQECCs) and obtain two new families of MDS EAQECCs with arbitrary parameters. [ABSTRACT FROM AUTHOR]

In this paper, we provide two complementary results for cyclic codes of composite length. First, we give a general construction of cyclic codes of length nr from cyclic codes of length n and a lower bound on the minimum distance. Numerical data show that many cyclic codes of composite length with the best parameters can be obtained in this way. Second, in the other direction, we show that for cyclic codes of length n with a primitive n-th root of unity as a non-zero, the minimum distance is roughly bounded by the square-free part of n. This means that we shall not expect good cyclic codes of length n in general if, for example, the length n is divisible by a high power of a prime. [ABSTRACT FROM AUTHOR]

Kim et al. (2021) gave a method to embed a given binary $[n,k]$ code $\mathcal {C}\,\,(k = 3, 4)$ into a self-orthogonal code of the shortest length which has the same dimension $k$ and minimum distance $d' \ge d(\mathcal {C})$. We extend this result by proposing a new method related to a special matrix, called the self-orthogonality matrix $SO_{k}$ , obtained by shortening a Reed-Muller code ${\mathcal R}(2,k)$. Using this approach, we can extend binary linear codes to many optimal self-orthogonal codes of dimensions 5 and 6. Furthermore, we partially disprove the conjecture (Kim et al. (2021)) by showing that if $31 \le n \le 256$ and $n\equiv 14,22,29 \pmod {31}$ , then there exist optimal $[n], [5]$ codes which are self-orthogonal. We also construct optimal self-orthogonal $[n], [6]$ codes when $41 \le n \le 256$ satisfies $n \ne 46, 54, 61$ and $n \equiv \!\!\!\!\!/~7, 14, 22, 29, 38, 45, 53, 60 \pmod {63}$. [ABSTRACT FROM AUTHOR]

Deep generative models have emerged as a powerful class of priors for signals in various inverse problems such as compressed sensing, phase retrieval and super-resolution. In this work, we consider the compressed sensing problem and assume the unknown signal to lie in the range of some pre-trained generative model. A popular approach for signal recovery is via gradient descent in the low-dimensional latent space. While gradient descent has achieved good empirical performance, its theoretical behavior is not well understood. We introduce the use of stochastic gradient Langevin dynamics (SGLD) for compressed sensing with a generative prior. Under mild assumptions on the generative model, we prove the convergence of SGLD to the true signal. We also demonstrate competitive empirical performance to standard gradient descent. [ABSTRACT FROM AUTHOR]

We introduce quantum pin codes: a class of quantum CSS codes. Quantum pin codes are a generalization of quantum color codes and Reed-Muller codes and share a lot of their structure and properties. Pin codes have gauge operators, an unfolding procedure and their stabilizers form so-called $\ell $ -orthogonal spaces meaning that the joint overlap between any $\ell $ stabilizer elements is always even. This last feature makes them interesting for devising magic-state distillation protocols, for instance by using puncturing techniques. We study examples of these codes and their properties. [ABSTRACT FROM AUTHOR]

We construct a family of $(n,k)$ convolutional codes with degree $\delta \in \{k,n-k\}$ that have a maximum distance profile. The field size required for our construction is $\Theta (n^{2\delta })$ , which improves upon the known constructions of convolutional codes with a maximum distance profile. Our construction is based on the theory of skew polynomials. [ABSTRACT FROM AUTHOR]

We study the Singleton-type bound that provides an upper limit on the minimum distance of locally repairable codes. We present an improved bound by carefully analyzing the combinatorial structure of the repair sets. Thus, we show the previous bound is unachievable for certain parameters. We then also provide explicit constructions of optimal codes which show that for certain parameters the new bound is sharp. Additionally, as a byproduct, some previously known codes are shown to attain the new bound and are thus proved to be optimal. [ABSTRACT FROM AUTHOR]

We study generalized covering radii, a fundamental property of linear codes that characterizes the trade-off between storage, latency, and access in linear data-query protocols such as PIR. We prove lower and upper bounds on the generalized covering radii of Reed-Muller codes, as well as finding their exact value in certain extreme cases. With the application to linear data-query protocols in mind, we also construct a covering algorithm that gets as input a set of points in space, and find a corresponding set of codewords from the Reed-Muller code that are jointly not farther away from the input than the upper bound on the generalized covering radius of the code. We prove that the algorithm runs in time that is polynomial in the code parameters. [ABSTRACT FROM AUTHOR]

This article discusses the security of McEliece-like encryption schemes using subspace subcodes of Reed–Solomon codes, i.e. subcodes of Reed–Solomon codes over ${\mathbb {F}_{q^{m}}}$ whose entries lie in a fixed collection of ${\mathbb {F}_{q}}$ –subspaces of ${\mathbb {F}_{q^{m}}}$. These codes appear to be a natural generalisation of Goppa and alternant codes and provide a broader flexibility in designing code based encryption schemes. For the security analysis, we introduce a new operation on codes called the twisted product which yields a polynomial time distinguisher on such subspace subcodes as soon as the chosen ${\mathbb {F}_{q}}$ –subspaces have dimension larger than $m/2$. From this distinguisher, we build an efficient attack which in particular breaks some parameters of a recent proposal due to Khathuria, Rosenthal and Weger. [ABSTRACT FROM AUTHOR]

Two algorithms for the classification of linear codes over finite fields are presented. One of the algorithms is based on canonical augmentation and the other one on lattice point enumeration. New classification results over fields with 2, 3 and 4 elements are obtained. [ABSTRACT FROM AUTHOR]

We investigate fat projective linear codes over ${\mathbb Z}_{p^{m}}$ , $m\geqslant 2$ , with two nonzero homogeneous weights (“two-weight codes”), building on the graph theory approach developed by Delsarte for codes over fields. Our main result is the classification of such codes under the additional assumption that the columns of a generator matrix of the code determine a cap in the projective Hjelmslev geometry $\mathop {\mathrm {PHG}}\nolimits (k-1, {\mathbb Z}_{p^{m}})$. This generalizes a result on projective two weight codes with dual distance at least four (Calderbank, 1982). The proof relies on a careful analysis of a certain strongly regular graph built on the cosets of the dual code, and on an interpretation of its parameters in terms of projective Hjelmslev geometry. [ABSTRACT FROM AUTHOR]

We consider a generalized construction of stream ciphers with forward security. The design framework is modular: it is built from a so-called layer function that updates the key and (optionally) the nonce and generates a new pseudorandom output stream. We analyze the generalized construction for four different instantiations: two possible layer functions that are in turn instantiated with either a block cipher or a pseudorandom function. We prove that each of these instantiations gives a stream cipher that is pseudorandom and forward secure in the multi-user setting with a very tight bound. A comprehensive analysis shows that the two block cipher based instantiations achieve very similar bounds. For the pseudorandom function based instantiations there is no clear winner: either layer can be beneficial over the other one, depending on the choice of parameters. By instantiating the pseudorandom function with a generic construction such as the sum of permutations, we obtain a highly efficient and competitive stream cipher based on an n-bit block cipher that is secure beyond the $2^{\text {n}/2}$ birthday bound. [ABSTRACT FROM AUTHOR]

As an important coding scheme in modern distributed storage systems, locally repairable codes (LRCs) have attracted a lot of attentions from perspectives of both practical applications and theoretical research. As a major topic in the research of LRCs, bounds and constructions of the corresponding optimal codes are of particular concerns. In this work, codes with $(r,\delta)$ -locality which have optimal minimal distance w.r.t. the bound given by Prakash et al. are considered. Through parity-check matrix approach, constructions of both optimal $(r,\delta)$ -LRCs with all symbol locality ($(r,\delta)_{a}$ -LRCs) and optimal $(r,\delta)$ -LRCs with information locality ($(r,\delta)_{i}$ -LRCs) are provided. As a generalization of a work of Xing and Yuan, these constructions are built on a connection between sparse hypergraphs and optimal $(r,\delta)$ -LRCs. With the help of constructions of large sparse hypergraphs, the lengths of codes obtained from our construction can be super-linear in the alphabet size. This improves upon previous constructions when the minimal distance of the code is at least $3\delta +1$. As two applications, optimal H-LRCs with super-linear lengths and GSD codes with unbounded lengths are also constructed. [ABSTRACT FROM AUTHOR]

We present a communication-efficient distributed protocol for computing the Babai point, an approximate nearest point for a random vector ${\mathbf{X}}\in \mathbb {R}^{n}$ in a given lattice. We show that the protocol is optimal in the sense that it minimizes the sum rate when the components of $\boldsymbol {X}$ are mutually independent. We then investigate the error probability, i.e. the probability that the Babai point does not coincide with the nearest lattice point, motivated by the fact that for some cases, a distributed algorithm for finding the Babai point is sufficient for finding the nearest lattice point itself. Two different probability models for $\boldsymbol {X}$ are considered—uniform and Gaussian. For the uniform model, in dimensions two and three, the error probability is seen to grow with the packing density, and we demonstrate that the densest lattice in dimension two presents the worst error probability. For higher dimensions, we develop probabilistic concentration bounds as well as bounds based on geometric arguments for the error probability. The probabilistic bounds lead to the conclusion that for lattices which generate suitably thin coverings of $\mathbb {R}^{n}$ (which includes lattices that meet Rogers’ bound on the covering radius), the error probability goes to unity as $n$ grows. Probabilistic and geometric bounds are also used to estimate the error probability under the uniform model for various lattices including the $A_{n}$ family and the Leech lattice, $\Lambda _{24}$. On the other hand, for the Gaussian model, the error probability goes to zero as the lattice dimension tends to infinity, provided the noise variance is sufficiently small. [ABSTRACT FROM AUTHOR]

Spectral bounds on the minimum distance of quasi-twisted codes over finite fields are proposed, based on eigenvalues of polynomial matrices and the corresponding eigenspaces. They generalize the Semenov-Trifonov and Zeh-Ling bounds in a way similar to how the Roos and shift bounds extend the BCH and HT bounds for cyclic codes. The eigencodes of a quasi-twisted code in the spectral theory and the outer codes in its concatenated structure are related. A comparison based on this relation verifies that the Jensen bound always outperforms the spectral bound under special conditions, which yields a similar relation between the Lally and the spectral bounds. The performances of the Lally, Jensen and spectral bounds are presented in comparison with each other. [ABSTRACT FROM AUTHOR]

Based on the applications to the wire-tap channel and bounds on the relative generalized Hamming weight, we will introduce optimal codes of type $\mathcal {I}$ and type $\mathcal {II}$. We will give some necessary conditions and the explicit construction for a linear code $\mathcal {C}$ to be optimal of type $\mathcal {I}$ and $\mathcal {II}$ with respect to a subcode $\mathcal {C}_{1}$. We will also present a new bound on two different parameters of the relative generalized Hamming weight. [ABSTRACT FROM AUTHOR]

In this work, cyclic-skew-cyclic codes and sum-rank BCH codes are introduced. Cyclic-skew-cyclic codes are characterized as left ideals of a suitable non-commutative finite ring, constructed using skew polynomials on top of polynomials (or vice versa). Single generators of such left ideals are found, and they are used to construct generator matrices of the corresponding codes. The notion of defining set is introduced, using pairs of roots of skew polynomials on top of poynomials. A lower bound (called sum-rank BCH bound) on the minimum sum-rank distance is given for cyclic-skew-cyclic codes whose defining set contains certain consecutive pairs. Sum-rank BCH codes, with prescribed minimum sum-rank distance, are then defined as the largest cyclic-skew-cyclic codes whose defining set contains such consecutive pairs. The defining set of a sum-rank BCH code is described, and a lower bound on its dimension is obtained. Thanks to it, tables are provided showing that sum-rank BCH codes beat previously known codes for the sum-rank metric for binary $2 \times 2 $ matrices (i.e., codes whose codewords are lists of $2 \times 2 $ binary matrices, for a wide range of list lengths that correspond to the code length). Finally, a decoder for sum-rank BCH codes up to half their prescribed distance is obtained. [ABSTRACT FROM AUTHOR]

Given a collection of probability distributions p1, . . . , pm, the minimum entropy coupling is the coupling X1, . . . , Xm (Xi ∼ pi) with the smallest entropy H(X1, . . . , Xm). While this problem is known to be NP-hard, we present an efficient algorithm for computing a coupling with entropy within 2 bits from the optimal value. More precisely, we construct a coupling with entropy within 2 bits from the entropy of the greatest lower bound of p1, . . . , pm with respect to majorization. This construction is also valid when the collection of distributions is infinite, and when the supports of the distributions are infinite. Potential applications of our results include random number generation, entropic causal inference, and functional representation of random variables. [ABSTRACT FROM AUTHOR]