Singleton bounds for R-additive codes

Abstract

Shiromoto (Linear Algebra Applic 295 (1999) 191-200) obtained the basic exact sequence for the Lee and Euclidean weights of linear codes over Z and as an application, he found the Singleton bounds for linear codes over Z with respect to Lee and Euclidean weights. Huffman (Adv. Math. Commun 7 (3) (2013) 349-378) obtained the Singleton bound for Fq-linear F q t-codes with respect to Hamming weight. Recently the theory of Fq-linear F q t-codes were generalized to R-additive codes over R-algebras by Samei and Mahmoudi. In this paper, we generalize Shiromoto's results for linear codes over Z to R-additive codes. As an application, when R is a chain ring, we obtain the Singleton bounds for R-additive codes over free R-algebras. Among other results, the Singleton bounds for additive codes over Galois rings are given.

Codes over $ \frak m $-adic completion rings

“…Codes meeting this bound are called M DR codes; see [19,Theorem 3.7]. Since the weight of the last row of the standard generator matrix of an m-adic code C of rank k is less than n − k + 1, d H (C) ≤ n − k + 1.…”

“…Recently the theory of F q -linear codes was generalized to R-additive codes [18,19]. When R is a finite chain ring, the structure of cyclic R-additive codes over free R-algebras has been given in [18].…”

Polynomial representation of additive cyclic codes and new quantum codes

“…Additive cyclic codes are of interest due to their rich algebraic properties and application in the construction of quantum codes. There have been several works in the literature toward the classification of additive cyclic codes for different applications [1,6,9,22,23,28], and also due to their connection to other families of block codes such as quasi-cyclic codes [21]. In [22], a canonical decomposition of additive cyclic code over F 4 was introduced using certain finite field extensions of F 4 .…”