Construction of Cyclic and Constacyclic Codes for b-symbol Read Channels Meeting the Plotkin-like Bound

The symbol-pair codes over finite fields have been raised for symbol-pair read channels and motivated by application of high-density data storage technologies [1, 2]. Their generalization is the code for b-symbol read channels (b > 2). Many MDS codes for b-symbol read channels have been constructed which meet the Singleton-like bound ([3, 4, 10] for b = 2 and [11] for b > 2). In this paper we show the Plotkin-like bound and present a construction on irreducible cyclic codes and constacyclic codes meeting the Plotkin-like bound

Generalized Pair Weights of Linear Codes and Linear Isomorphisms Preserving Pair Weights

In this paper, we first introduce the notion of generalized pair weights of an $[n, k]$-linear code over the finite field $\mathbb{F}_q$ and the notion of pair $r$-equiweight codes, where $1\le r\le k-1$. Some basic properties of generalized pair weights of linear codes over finite fields are derived. Then we obtain a necessary and sufficient condition for an $[n,k]$-linear code to be a pair equiweight code, and we characterize pair $r$-equiweight codes for any $1\le r\le k-1$. Finally, a necessary and sufficient condition for a linear isomorphism preserving pair weights between two linear codes is obtained

Generalized bb-weights and bb-MDS Codes

In this paper, we first introduce the notion of generalized $b$-weights of $[n,k]$-linear codes over finite fields, and obtain some basic properties and bounds of generalized $b$-weights of linear codes which is called Singleton bound for generalized $b$-weights in this paper. Then we obtain a necessary and sufficient condition for an $[n,k]$-linear code to be a $b$-MDS code by using generator matrixes of this linear code and parity check matrixes of this linear code respectively. Next a theorem of a necessary and sufficient condition for a linear isomorphism preserving $b$-weights between two linear codes is obtained, in particular when $b=1$, this theorem is the MacWilliams extension theorem. Then we give a reduction theorem for the MDS conjecture. Finally, we calculate the generalized $b$-weight matrix $D(C)$ when $C$ is simplex codes or two especial Hamming codes