Narrow Results

In a previous paper, the second and third named authors introduced the concept of the complete cycle index and discussed a relation with the complete weight enumerator in coding theory. In this paper, we introduce the concept of the complete joint cycle index and the average complete joint cycle index, and discuss a relation with the complete joint weight enumerator and the average complete joint weight enumerator, respectively, in coding theory. Moreover, the notion of the average intersection numbers is given, and we discuss a relation with the average intersection numbers in coding theory.

In this work, we introduce a new class of optimal tensor codes related to the Ravagnani-type anticodes defined in 2023. We show that it extends the family of $j$-maximum rank distance codes and contains the $j$-binomial moment determined codes (with respect to the Ravagnani-type anticodes) as a proper subclass. We define and study the zeta function for tensor codes. We establish connections between this object and the weight enumerator of a tensor code with respect to the Ravagnani-type anticodes. We introduce a new refinement of the invariants of tensor codes exploiting the structure of product lattices of some classes of anticodes classified in 2023 and we derive the corresponding MacWilliams identities. In this framework, we also define a multivariate version of the tensor weight enumerator and we establish relations with the corresponding zeta function. As an application, we derive connections on the tensor weights.

The Eisenstein polynomial is the weighted sum of the weight enumerators of all classes of Type II codes of fixed length. In this note, we investigate the ring generated by Eisenstein polynomials in genus 2.

There is a limited class of perfect codes with respect to the classical Hamming metric. There are other kind of metrics with respect to which perfect codes have been investigated viz. poset metric, block metric and poset block metric. Given the minimal elements of a poset, a necessary and sufficient condition for $1$-perfectness of a poset block code has been derived. A necessary and sufficient condition for a poset block code to be $r$-perfect has also been considered. Further, for each $r$, $1\le r\le n-k$, a sufficient condition that ensures the existence of a poset block structure which turns a given code into an $r$-perfect poset block code has been obtained. Several illustrations of well known codes to be $r$-perfect for specific values of $r$ have been explored.

Self-dual and maximal self-orthogonal codes over $\mathrm{\text{GF}}(q)$, where $q$ is even or $q\equiv 1$(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 $W_{\overline{C}}(x,y)$ of a self-dual code $\overline{C}$ is given in terms of a maximal self-orthogonal code $C$, where $\overline{C}$ is obtained by the extension of $C$.

In the last 60 years coding theory has been studied a lot over finite fields ${𝔽}_q$ or commutative rings ${ℛ}$ with unity. Although in $1993$, a study on the classification of the rings (not necessarily commutative or ring with unity) of order $p^2$ had been presented, the construction of codes over non-commutative rings or non-commutative non-unital rings surfaced merely two years ago. In this letter, we extend the diverse research on exploring the codes over the non-commutative and non-unital ring $E=〈2a=2b=0,a^2=a,b^2=b,ab=a,ba=b〉$ by presenting the classification of optimal and nice codes of length $n\le 7$ over $E$, along with respective weight enumerators and complete weight enumerators.

Linear codes over finite chain rings correspond to multisets of points in finite projective Hjelmslev geometries. This survey contains an introduction to both subjects, working out the links between coding theory and geometry over this class of rings.