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We obtain a characterization on self-orthogonality for a given binary linear code in terms of the number of column vectors in its generator matrix, which extends the result of Bouyukliev et al. (2006). As an application, we give an algorithmic method to embed a given binary k-dimensional linear code C (k = 3,4) into a self-orthogonal code of the shortest length which has the same dimension k and minimum distance d' ≥ d(C). For k > 4, we suggest a recursive method to embed a k -dimensional linear code to a self-orthogonal code. We also give new explicit formulas for the minimum distances of optimal self-orthogonal codes for any length n with dimension 4 and any length n ≢ 6, 13,14,21,22,28,29 (mod 31) with dimension 5. [ABSTRACT FROM AUTHOR]

The weight spectra of MDS codes of length n and dimension k over arbitrary alphabets are studied. For all q-ary MDS codes of dimension k and length ${n}\ne {q}+{k}-1 $ containing the zero codeword, it is shown that all k weights from n to ${n}-{k}+1 $ are realized. The weight spectrum in the remaining case ${n}={q}+{k}-1 $ is also determined. Additionally, we prove that all binary MDS codes are equivalent to linear MDS codes. The proofs are combinatorial, and self contained. [ABSTRACT FROM AUTHOR]

An additive quaternary [n,k,d]-code (length n, quaternary dimension k, minimum distance d) is a 2k-dimensional F2-vector space of n-tuples with entries in F2 ⊕ F2 (the 2-dimensional vector space over F2) with minimum Hamming distance d. We determine the optimal parameters of additive quaternary codes of dimension k ≤ 3. The most challenging case is dimension k = 2.5. We prove that an additive quaternary [n,2.5,d]-code where d < n−1 exists if and only if 3(n−d) ≥ ⌈d/2⌉ + ⌈d/4⌉ + ⌈d/8⌉. In particular, we construct new optimal 2.5-dimensional additive quaternary codes. As a by-product, we give a direct proof for the fact that a binary linear [3m,5,2e]2-code for e < m−1 exists if and only if the Griesmer bound 3(m−e) ≥e/2⌉ + ⌈e/4⌉ + ⌈e/8⌉ is satisfied. [ABSTRACT FROM AUTHOR]

Recently, some infinite families of binary minimal and optimal linear codes were constructed from simplicial complexes by Hyun et al. Inspired by their work, we present two new constructions of codes over the ring $\mathbb {F}_{2}+u\mathbb {F}_{2}$ by employing simplicial complexes. When the simplicial complexes are all generated by a maximal element, we determine the Lee weight distributions of two classes of the codes over $\mathbb {F}_{2}+u\mathbb {F}_{2}$. Our results show that the codes have few Lee weights. Via the Gray map, we obtain an infinite family of binary codes meeting the Griesmer bound and a class of binary distance optimal codes. [ABSTRACT FROM AUTHOR]

This paper investigates fundamental properties of nonlinear binary codes by looking at the codebook matrix not row-wise (codewords), but column-wise. The family of weak flip codes is presented and shown to contain many beautiful properties. In particular the subfamily fair weak flip codes, which goes back to Shannon et al. and which was shown to achieve the error exponent with a fixed number of codewords $\mathsf {M}$ , can be seen as a generalization of linear codes to an arbitrary number of codewords. The fair weak flip codes are related to binary nonlinear Hadamard codes. Based on the column-wise approach to the codebook matrix, the $r$ -wise Hamming distance is introduced as a generalization to the well-known and widely used (pairwise) Hamming distance. It is shown that the minimum $r$ -wise Hamming distance satisfies a generalized $r$ -wise Plotkin bound. The $r$ -wise Hamming distance structure of the nonlinear fair weak flip codes is analyzed and shown to be superior to many codes. In particular, it is proven that the fair weak flip codes achieve the $r$ -wise Plotkin bound with equality for all $r$. In the second part of this paper, these insights are applied to a binary erasure channel with an arbitrary erasure probability 0 < $\delta$ < 1. An exact formula for the average error probability of an arbitrary (linear or nonlinear) code using maximum likelihood decoding is derived and shown to be expressible using only the $r$ -wise Hamming distance structure of the code. For a number of codewords $\mathsf {M}$ satisfying $\mathsf {M}\le 4$ and an arbitrary finite blocklength $n$ , the globally optimal codes (in the sense of minimizing the average error probability) are found. For $\mathsf {M}$ = 5 or $\mathsf {M}$ = 6 and an arbitrary finite blocklength $n$ , the optimal codes are conjectured. For larger $\mathsf {M}$ , observations regarding the optimal design are presented, e.g., that good codes have a large $r$ -wise Hamming distance structure for all $r$. Numerical results validate our code design criteria and show the superiority of our best found nonlinear weak flip codes compared with the best linear codes. [ABSTRACT FROM AUTHOR]

Codes over Galois rings have been studied extensively during the last three decades. Negacyclic codes over \mathop \mathrm GR\nolimits (2^a,m) of length 2^s have been characterized: the ring \mathcal R2(a,m,-1)= {\mathrm{ GR}}(2^{a},m)[x]/\langle x^{2^{s}}+1\rangle is a chain ring. Furthermore, these results have been generalized to \lambda -constacyclic codes for any unit \lambda of the form 4z-1 , . In this paper, we study more general cases and investigate all cases, where {\mathcal{ R}}_{p}(a,m,\gamma) = {\mathrm{ GR}}(p^{a},m)[x]/\langle x^{p^{s}}-\gamma \rangle is a chain ring. In particular, the necessary and sufficient conditions for the ring {\mathcal{ R}}_{p}(a,m,\gamma) to be a chain ring are obtained. In addition, by using this structure we investigate all \gamma -constacyclic codes over \mathop {\mathrm {GR}}\nolimits (p^{a},m) when {\mathcal{ R}}_{p}(a,m,\gamma)$ is a chain ring. The necessary and sufficient conditions for the existence of self-orthogonal and self-dual \gamma$ -constacyclic codes are also provided. Among others, for any prime p$ , the structure of is used to establish the Hamming and homogeneous distances of $\gamma$ -constacyclic codes. [ABSTRACT FROM PUBLISHER]

In this paper, a generic construction of codebooks based on binary codes is introduced. With this generic construction, a few previous constructions of optimal codebooks are extended, and a new class of codebooks almost meeting the Levenshtein bound is presented. Exponentially many codebooks meeting or almost meeting the Levenshtein bound from binary codes are obtained in this paper. The codebooks constructed in this paper have alphabet size 4. As a byproduct, three bounds on the parameters of binary codes are derived. [ABSTRACT FROM PUBLISHER]

We present new constructions of codes for asymmetric channels for both binary and nonbinary alphabets, based on methods of generalized code concatenation. For the binary asymmetric channel, our methods construct nonlinear single-error-correcting codes from ternary outer codes. We show that some of the Varshamov–Tenengol’ts–Constantin–Rao codes, a class of binary nonlinear codes for this channel, have a nice structure when viewed as ternary codes. In many cases, our ternary construction yields even better codes. For the nonbinary asymmetric channel, our methods construct linear codes for many lengths and distances which are superior to the linear codes of the same length capable of correcting the same number of symmetric errors. [ABSTRACT FROM AUTHOR]

We study certain combinatorial aspects of list-decoding, motivated by the exponential gap between the known upper bound (of \(O(1/\gamma )\) ) and lower bound (of \(\Omega _{p} (\log (1/\gamma ))\) ) for the list size needed to list decode up to error fraction \(p\) with rate \(\gamma \) away from capacity, i.e., \(1- h(p)-\gamma \) [here \(p\in (0, {1}/{2})\) and \(\gamma > 0\) ]. Our main result is that we prove that in any binary code \(C \subseteq \{ 0, 1 \} ^{n}\) of rate \(1- h(p) - \gamma \) , there must exist a set \( \mathcal {L}\subset C\) of \(\Omega _{p} (1/\sqrt {\gamma })\) codewords such that the average distance of the points in \( \mathcal {L}\) from their centroid is at most \(pn\) . In other words, there must exist \(\Omega _{p}(1/\sqrt {\gamma })\) codewords with low average radius. The standard notion of list decoding corresponds to working with the maximum distance of a collection of codewords from a center instead of average distance. The average radius form is in itself quite natural; for instance, the classical Johnson bound in fact implies average-radius list-decodability. The remaining results concern the standard notion of list-decoding, and help clarify the current state of affairs regarding combinatorial bounds for list-decoding as follows. First, we give a short simple proof, over all fixed alphabets, of the above-mentioned \(\Omega _{p}(\log (1/\gamma ))\) lower bound. Earlier, this bound followed from a complicated, more general result of Blinovsky. Second, we show that one cannot improve the \(\Omega _{p}(\log (1/\gamma ))\) lower bound via techniques based on identifying the zero-rate regime for list-decoding of constant-weight codes [this is a typical approach for negative results in coding theory, including the \(\Omega _{p} (\log (1/\gamma ))\) list-size lower bound]. On a positive note, our \(\Omega _{p}(1/\sqrt {\gamma })\) lower bound for average-radius list-decoding circumvents this barrier. Third, we exhibit a reverse connection between the existence of constant-weight and general codes for list-decoding, showing that the best possible list-size, as a function of the gap \(\gamma \) of the rate to the capacity limit, is the same up to constant factors for both constant-weight codes (with weight bounded away from \(p\) ) and general codes. Fourth, we give simple second moment-based proofs that w.h.p. a list-size of \(\Omega _{p} (1/\gamma )\) is needed for list-decoding random codes from errors as well as erasures. For random linear codes, the corresponding list-size bounds are \(\Omega _{p} (1/\gamma )\) for errors and \(\exp (\Omega _{p}(1/\gamma ))\) for erasures. [ABSTRACT FROM AUTHOR]