Correction: Hamming weight enumerators and hulls of linear codes with new parameters associated with simplicial complexes

Hamming weight enumerators and hulls of linear codes with new parameters associated with simplicial complexes

In 1999, Iwan Duursma defined the zeta function for a linear code as a generating function of its Hamming weight enumerator. It has various properties similar to those of the zeta function of an algebraic curve. This article extends Duursma's theory to the case of formal weight enumerators. It is shown that the zeta function for a formal weight enumerator has a similar structure to that of the weight enumerator of a Type II code. The notion of the extremal formal weight enumerators is introduced and an analogue of the Mallows-Sloane bound is obtained. Moreover the ternary case is considered.

Self-dual codes over the integers modulo 4

On a Riemann hypothesis analogue for selfdual weight enumerators of genus less than 3

This paper investigates the extended weight enumerators for the number-theoretic insertion/deletion correcting codes. As a special case, this paper provides the Hamming weight enumerators and cardinalities of the Tenengolts’ non-binary codes, which are non-binary codes correcting single insertion/deletion.

The purpose of this paper is to improve the upper bounds of the minimum distances of self-dual codes over $${\mathbb{F}}_5$$ for lengths [22, 26, 28, 32---40]. In particular, we prove that there is no [22, 11, 9] self-dual code over $${\mathbb{F}}_5$$ , whose existence was left open in 1982. We also show that both the Hamming weight enumerator and the Lee weight enumerator of a putative [24, 12, 10] self-dual code over $${\mathbb{F}}_5$$ are unique. Using the building-up construction, we show that there are exactly nine inequivalent optimal self-dual [18, 9, 7] codes over $${\mathbb{F}}_5$$ up to the monomial equivalence, and construct one new optimal self-dual [20, 10, 8] code over $${\mathbb{F}}_5$$ and at least 40 new inequivalent optimal self-dual [22, 11, 8] codes.

In this paper, we study a type of mixed alphabet codes, called [Formula: see text]-linear codes, where [Formula: see text] with [Formula: see text]. First, we define and discuss several weight enumerators of [Formula: see text]-linear codes, including Hamming weight, Lee weight and symmetrized weight enumerators. MacWilliams identity connects the weight enumerator of codes and their dual codes. Further, we present the MacWilliams identities about Lee weight and complete weight over the ring [Formula: see text]. Finally, several examples are presented to illustrate our main results.

Construction of several infinite families of linear codes with new parameters: Hamming weight enumerators and hull dimensions

An abundance of invariant polynomials satisfying the Riemann hypothesis

On the Hamming Weight Enumerators of Self-Dual Codes over [formula omitted

In this article we introduce a method of constructing binary linear codes and computing their weights by means of Boolean functions arising from mathematical objects called simplicial complexes. Inspired by Adamaszek (Am Math Mon 122:367–370, 2015) we introduce n-variable generating functions associated with simplicial complexes and derive explicit formulae. Applying the construction (Carlet in Finite Field Appl 13:121–135, 2007; Wadayama in Des Codes Cryptogr 23:23–33, 2001) of binary linear codes to Boolean functions arising from simplicial complexes, we obtain a class of optimal linear codes and a class of minimal linear codes.

A linear code is optimal if it has the highest minimum distance of any linear code with a given length and dimension. We construct infinite families of optimal binary linear codes C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δc</sub> constructed from simplicial complexes in F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , where Δ is a simplicial complex in F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> and Δ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sup> the complement of Δ. We first find an explicit computable criterion for C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δc</sub> to be optimal; this criterion is given in terms of the 2-adic valuation of Σ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j=1</sub> 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|Ai|-1</sup> , where the At's are maximal elements of Δ. Furthermore, we obtain much simpler criteria under various specific conditions on the maximal elements of Δ. In particular, we find that C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δc</sub> is a Griesmer code if and only if the maximal elements of Δ are pairwise disjoint and their sizes are all distinct. Specially, when f has exactly two maximal elements, we explicitly determine the weight distribution of C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δc</sub> .We present many optimal linear codes constructed by our method, and we emphasize that we obtain at least 32 new optimal linear codes.

Self-dual and maximal self-orthogonal codes over [Formula: see text], where [Formula: see text] is even or [Formula: see text](mod 4), are extensively studied in this paper. We prove that every maximal self-orthogonal code can be extended to a self-dual code as in the case of binary Golay code. Using these results, we show that a self-dual code can also be constructed by gluing theory even if the sum of the lengths of the gluing components is odd. Furthermore, the (Hamming) weight enumerator [Formula: see text] of a self-dual code [Formula: see text] is given in terms of a maximal self-orthogonal code [Formula: see text], where [Formula: see text] is obtained by the extension of [Formula: see text].

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 F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> + uF <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> 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 F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> + uF <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> . 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.

The Hamming and complete weight enumerators of maximally self-orthogonal codes over $GF$(3) of lengths $12m - 1$, $12m$ and $12m + 1$ are characterized. The results for length $12m + 1$ are believed to be new, while those for length $12m - 1$, and $12m$ have been considerably simplified.