Linear codes from simplicial complexes

<p style='text-indent:20px;'>Recently, some infinite families of optimal binary linear codes are constructed from simplicial complexes. Afterwards, the construction method was extended to using arbitrary posets. In this paper, based on a generic construction of linear codes, we obtain four classes of optimal binary linear codes by using the posets of two chains. Two of them induce Griesmer codes which are not equivalent to the linear codes constructed by Belov. Those codes are exploited to construct secret sharing schemes in cryptography as well.</p>

As a special type of linear codes, minimal linear codes have important applications in secret sharing. Up to now, only a few infinite families of minimal binary linear codes with $$w_{\min }/w_{\max }\le 1/2$$ were reported in the literature, while vast knowledge exists on the ones with $$w_{\min }/w_{\max }> 1/2$$ . Herein, $$w_{\min }$$ and $$w_{\max }$$ respectively denote the minimum and maximum nonzero Hamming weights in a linear code. Recently, several classes of linear codes with certain properties were constructed by Zhou et al. from a generic construction. The objective of this paper is to obtain four families of minimal binary linear codes with $$w_{\min }/w_{\max }\le 1/2$$ from those linear codes proposed by Zhou et al. The parameters of our minimal linear codes are quite different from known ones. Based on the properties of Krawtchouk polynomials, the weight distributions of all these four families of binary linear codes are established.

This paper investigates the structures and properties of one-Lee weight codes and two-Lee weight projective codes over ℤ4. The authors first give the Pless identities on the Lee weight of linear codes over ℤ4. Then the authors study the necessary conditions for linear codes to have one-Lee weight and two-Lee projective weight respectively, the construction methods of one-Lee weight and two-Lee weight projective codes over ℤ4 are also given. Finally, the authors recall the weight-preserving Gray map from (ℤ 4 , Lee weight) to ( $\mathbb{F}_2^{2n} $ , Hamming weight), and produce a family of binary optimal oneweight linear codes and a family of optimal binary two-weight projective linear codes, which reach the Plotkin bound and the Griesmer bound.

This paper studies linear and affine error-correcting codes for correcting synchronization errors such as insertions and deletions. We call such codes linear/affine insdel codes. Linear codes that can correct even a single deletion are limited to having an information rate at most (achieved by the trivial two fold repetition code). Previously, it was (erroneously) reported that more generally no nontrivial linear codes correcting deletions exist, i.e., that the -fold repetition codes and its rate of are basically optimal for any . We disprove this and show the existence of binary linear codes of length and rate just below capable of correcting insertions and deletions. This identifies rate as a sharp threshold for recovery from deletions for linear codes and reopens the quest for a better understanding of the capabilities of linear codes for correcting insertions/deletions. We prove novel outer bounds and existential inner bounds for the rate vs. (edit) distance trade-off of linear insdel codes. We complement our existential results with an efficient synchronization-string-based transformation that converts any asymptotically good linear code for Hamming errors into an asymptotically good linear code for insdel errors. Last, we show that the -rate limitation does not hold for affine codes by giving an explicit affine code of rate which can efficiently correct a constant fraction of insdel errors.

Deep holes are very important in the decoding of generalized RS codes, and deep holes of RS codes have been widely studied, but there are few works on constructing general linear codes based on deep holes. Therefore, we consider constructing binary linear codes by combining deep holes with binary BCH codes. In this article, we consider the 2-error-correcting binary primitive BCH codes and the extended codes to construct new binary linear codes by combining them with deep holes, respectively. Furthermore, three classes of binary linear codes are constructed, and then we determine the parameters and the weight distributions of these new binary linear codes.

Subfield codes of CD-codes over [formula omitted]

Several families of [formula omitted]-ary minimal linear codes with [formula omitted

Nonlinear channel codes are employed to protect two information fields that are related to the daylight saving time (DST) and leap second notifications within the enhanced WWVB broadcast format. The general codebook design criteria and the construction of these codes using integer linear program (ILP) is formulated. Specifically, two different variations are presented: i) minimizing the union bound of probability of word error (PWE), which is a tighter bound but has high computational complexity and ii) minimizing an analytical closed form approximation to the PWE, which has much lower complexity but is a looser bound on the exact PWE. By leveraging the highly non-uniform a-priori probabilities of the different messages being transmitted in these two fields, the proposed nonlinear codes provide coding gains of about 2–3 dB in the presence of additive white Gaussian noise (AWGN). When compared to checksum codes with the same rate, the nonlinear codes offer performance advantages of about 1–2 dB for the DST notification field. In the presence of impulsive noise, simulations and analysis show that the probability of word error when employing the proposed nonlinear code is lower by many orders of magnitude compared with that of the uncoded system.

A strategy is described which allowed new optimal binary linear codes to be constructed. Using the method, among other codes a new [48, 16, 16] optimal linear binary block code was constructed. The construction of this new code settles a conjecture mentioned in the paper by Janosov and colleagues (see appropriate Reference in Letter) where the authors found a new optimal [47, 15, 16]‐code, whose parameters are also relevant to the present author's new method.

The parameters of minimal linear codes

It has been well-known that the class of quasi-cyclic (QC) codes contain many good codes. In thispaper, a method to conduct a computer search for binary 2-generator QC codes is presented, anda large number of good 2-generator QC codes have been obtained. 5 new binary QC codes thatimprove the lower bounds on minimum distance are presented. Furthermore, with new 2-generatorQC codes and Construction X, 2 new improved binary linear codes are obtained. With the standardconstruction techniques, another 16 new binary linear codes that improve the lower bound on theminimum distance have also been obtained.

A code $${{\mathcal C}}$$ is $${{\mathbb{Z}_2\mathbb{Z}_4}}$$ -additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of $${{\mathcal C}}$$ by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of $${{\mathbb{Z}_2\mathbb{Z}_4}}$$ -additive codes under an extended Gray map are called $${{\mathbb{Z}_2\mathbb{Z}_4}}$$ -linear codes. In this paper, the invariants for $${{\mathbb{Z}_2\mathbb{Z}_4}}$$ -linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of $${{\mathbb{Z}_2\mathbb{Z}_4}}$$ -linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a $${{\mathbb{Z}_2\mathbb{Z}_4}}$$ -linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a $${{\mathbb{Z}_2\mathbb{Z}_4}}$$ -linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a $${{\mathbb{Z}_2\mathbb{Z}_4}}$$ -linear code for each possible pair (r, k) is given.

A new family of nonlinear codes, called weak flip codes, is presented and is shown to contain many beautiful properties. In particular, the subfamily of fair weak flip codes can be seen as a generalization of linear codes. Different from linear codes that only exist for a number of codewords M being an integer-power of 2, the fair weak flip code can be defined for an arbitrary M. It is then noted that the fair weak flip codes are related to binary nonlinear Hadamard codes: both code families maximize the minimum Hamming distance and meet the Plotkin bound. However, while the binary nonlinear Hadamard codes have only been shown to possess good Hamming-distance properties, the fair weak flip codes are proven to be globally optimal (in the sense of minimizing the error probability) among all linear or nonlinear codes for the binary erasure channel (BEC) for many values of the blocklength n and for M≤6. For M>6, similar optimality results are conjectured. The results in this paper are founded on a new powerful tool for the analysis and generation of block codes: the column-wise approach to the codebook matrix.

Consider a maximum-length shift-register sequence generated by a primitive polynomial f over a finite field. The set of its subintervals is a linear code whose dual code is formed by all polynomials divisible by f. Since the minimum weight of dual codes is directly related to the strength of the corresponding orthogonal arrays, we can produce orthogonal arrays by studying divisibility of polynomials. Munemasa (Finite Fields Appl 4(3):252---260, 1998) uses trinomials over $${\mathbb{F}_2}$$ to construct orthogonal arrays of guaranteed strength 2 (and almost strength 3). That result was extended by Dewar et al. (Des Codes Cryptogr 45:1---17, 2007) to construct orthogonal arrays of guaranteed strength 3 by considering divisibility of trinomials by pentanomials over $${\mathbb{F}_2}$$ . Here we first simplify the requirement in Munemasa's approach that the characteristic polynomial of the sequence must be primitive: we show that the method applies even to the much broader class of polynomials with no repeated roots. Then we give characterizations of divisibility for binomials and trinomials over $${\mathbb{F}_3}$$ . Some of our results apply to any finite field $${\mathbb{F}_q}$$ with q elements.

Determining the insdel distance of linear codes is a very challenging problem. Recently, Ji et al. [‘Strict half-Singleton bound, strict direct upper bound for linear insertion-deletion codes and optimal codes’, IEEE Trans. Inform. Theory 69 (5) (2023), 2900–2910] proposed their strict half-Singleton bound for linear codes that do not include the all-one vector. A natural question asks for linear codes with both large Hamming distance and insdel distance. Almost maximum distance separable (MDS) codes are a special kind of linear code with good Hamming error-correcting capability. We present a sufficient condition for the insdel distance of almost MDS codes to be bounded by the strict half-Singleton bound. In addition, we construct a class of two-dimensional near MDS codes that achieve the strict half-Singleton bound using twisted Reed–Solomon codes. Finally, we present a construction of optimal almost MDS insdel codes from Reed–Solomon codes for large dimensions.