Lexicodes over Rings

Designs, Codes and Cryptography, 2013

Codes over an infinite family of rings which are an extension of the binary field are defined. Two Gray maps to the binary field are attached and are shown to be conjugate. Euclidean and Hermitian self-dual codes are related to binary self-dual and formally self-dual codes, giving a construction of formally self-dual codes from a collection of arbitrary binary codes. We relate codes over these rings to complex lattices. A Singleton bound is proved for these codes with respect to the Lee weight. The structure of cyclic codes and their Gray image is studied. Infinite families of self-dual and formally self-dual quasi-cyclic codes are constructed from these codes.

Cryptography and Communications, 2022

In this paper we give the generalization of lifted codes over any finite chain ring. This has been done by using the construction of finite chain rings from p-adic fields. Further we propose a lattice construction from linear codes over finite chain rings using lifted codes. Codes over rings • Lifted codes • Lattices 1 Introduction Codes over finite rings have received significant attention in recent decades. Several authors have studied these codes due to their relationship with lattices construction, among other properties. The class of p-adic codes was introduced in [1]. Calderbank and Sloane investigated codes over p-adic integers and studied lifts of codes over F p and Z p e . Lifted codes over finite chain rings were studied in , however this study was restricted to the finite chain rings of the form F q [t]/ t k as pointed out by the reviewer in . Later, Young Ho generalized the construction of cyclic lifted codes for arbitrary finite fields to codes over Galois rings GR(p e , r) in . In , the reviewer stated that a unified treatment valid for all chain rings would certainly be desirable. Therefore, this study investigates the structure of finite chain rings as non-trivial quotient of ring integers of p-adic fields to generalize the construction in . A finite commutative chain ring is a finite local ring whose maximal ideals are principal. Any finite chain ring can be constructed from p-adic fields (see for

Journal of Algebra Combinatorics Discrete Structures and Applications

We describe a bordered construction for self-dual codes coming from group rings. We apply the constructions coming from the cyclic and dihedral groups over several alphabets to obtain extremal binary self-dual codes of various lengths. In particular we find a new extremal binary self-dual code of length 78.

Bulletin of the Korean Mathematical Society, 2015

This paper determines the structures of one-homogeneous weight codes over finite chain rings and studies the algebraic properties of these codes. We present explicit constructions of one-homogeneous weight codes over finite chain rings. By taking advantage of the distancepreserving Gray map defined in [7] from the finite chain ring to its residue field, we obtain a family of optimal one-Hamming weight codes over the residue field. Further, we propose a generalized method that also includes the examples of optimal codes obtained by Shi et al. in [17].

Advances in Mathematics of Communications, 2019

We describe eight composite constructions from group rings where the orders of the groups are 4 and 8, which are then applied to find self-dual codes of length 16 over F 4. These codes have binary images with parameters [32, 16, 8] or [32, 16, 6]. These are lifted to codes over F 4 + uF 4 , to obtain codes with Gray images of extremal self-dual binary codes of length 64. Finally, we use a building-up method over F 2 +uF 2 to obtain new extremal binary self-dual codes of length 68. We construct 11 new codes via the building-up method and 2 new codes by considering possible neighbors.

Siam Journal on Discrete Mathematics, 2006

The structure of multivariate semisimple codes over a finite chain ring R is established using the structure of the residue fieldR. Multivariate codes extend in a natural way the univariate cyclic and negacyclic codes and include some non-trivial codes over R. The structure of the dual codes in the semisimple abelian case is also derived and some conditions on the existence of selfdual codes over R are studied.

Information Theory, …, 1999

Cornell University - arXiv, 2022

In the last 60 years coding theory has been studied a lot over finite fields Fq 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 ≤ 7 over E, along-with respective weight enumerators and complete weight enumerators.

Advances in Mathematics of Communications, 2019

Galois images of polycyclic codes over a finite chain ring S and their annihilator dual are investigated. The case when a polycyclic codes is Galois-disjoint over the ring S, is characterized and, the trace codes and restrictions of free polycyclic codes over S are also determined givind an analogue of Delsarte theorem among trace map, any S-linear code and its annihilator dual.

Finite Fields and Their Applications

We introduce a bordered construction over group rings for self-dual codes. We apply the constructions over the binary field and the ring F 2 + uF 2 , using groups of

arXiv (Cornell University), 2023

In this article, we construct linear codes over the commutative non-unital ring I of size four. We obtain their Lee-weight distributions and study their binary Gray images. Under certain mild conditions, these classes of binary codes are minimal and self-orthogonal. All codes in this article are few-weight codes. Besides, an infinite class of these binary codes consists of distance optimal codes with respect to the Griesmer bound.

arXiv (Cornell University), 2019

Linear complementary dual (LCD) codes and linear complementary pair (LCP) of codes over finite fields have been intensively studied recently due to their applications in cryptography, in the context of side channel and fault injection attacks. The security parameter for an LCP of codes (C, D) is defined as the minimum of the minimum distances d(C) and d(D ⊥). It has been recently shown that if C and D are both 2-sided group codes over a finite field, then C and D ⊥ are permutation equivalent. Hence the security parameter for an LCP of 2-sided group codes (C, D) is simply d(C). We extend this result to 2-sided group codes over finite chain rings.

Applicable Algebra in Engineering, Communication and Computing, 2004

It is known that if a finite ring R is Frobenius then equivalences of linear codes over R are always monomial transformations. Among other results, in this paper we show that the converse of this result holds for finite local and homogeneous semilocal rings. Namely, it is shown that for every finite ring R which is a direct sum of local and homogeneous semilocal subrings, if every Hamming-weight preserving R-linear transformation of a codeC 1 onto a code C 2 is a monomial transformation then R is a Frobenius ring.

Discrete Mathematics, 1997

It is well known that cyclic linear codes of length n over a (finite) field F can be characterized in terms of the factors of the polynomial x"-1 in F[x]. This paper investigates cyclic linear codes over arbitrary (not necessarily commutative) finite tings and proves the above characterization to be true for a large class of such codes over these rings. (~