On the b-symbol weights of linear codes for large b

Abstract

Recently, b-symbol metric linear codes, which are extensions of linear codes with Hamming weight and symbol-pair weight, have attracted much attention due to their repair efficiency in b-symbol read channels. Extensive efforts have been dedicated to studying the classic Hamming weight and symbol-pair weight of linear codes, but there has been relatively less progress regarding the general b-symbol weight for $b\ge 3$. In this paper, we mainly consider the b-symbol weight of linear codes for large b. Firstly, we establish a direct correspondence between the b-symbol weight distribution of a linear code and the Hamming weight distribution of another linear code. Using this relation, we derive a Griesmer-type bound on the minimum b-symbol weight, which partially settles a conjecture of Shi, Zhu and Helleseth. Furthermore, we discover that the minimum b-symbol distance of a $[n,k]$ cyclic code is equal to the minimum b-symbol distance of certain linear code of parameters $[n,k-b]$ and prove that every $[n,k]$ cyclic code is always b-symbol MDS for $b\ge \min (n-k,k+1-\lceil \frac{d^{\perp }}{2}\rceil )$, where $d^{\perp }$ is the minimum distance of its dual. When $b\ge k+1-\lceil \frac{d^{\perp }}{2}\rceil$ we also determine its b-symbol weight distribution. Finally, we investigate the minimum b-symbol weight of several specific cyclic codes, including QR codes, irreducible cyclic codes, Kasami codes and the duals of double-error-correcting BCH codes.