MDS Constacyclic Codes and MDS Symbol-Pair Constacyclic Codes

Abstract

Symbol-pair codes are used to protect against symbol-pair errors in high density data storage systems. One of the most important tasks in symbol-pair coding theory is to design MDS codes. MDS symbol-pair codes are optimal in the sense that such codes attain the Singleton bound. In this paper, a new class of MDS symbol-pair codes with code-length 5p and optimal pair distance of 7 is established. It is shown that for any prime p ≡ 1 (mod 5), we can always construct four p-ary MDS symbol-pair cyclic codes of length 5p of largest possible pair distance 7. We also completely determined all MDS symbol-pair and MDS b-symbol codes of length p s and 2p s over F p m + uF p m by filling in some missing cases, and rectifying some gaps in Type 3 codes of recent papers. As an applications of our results, we use MAGMA to provide many examples of new MDS codes over F p m and F p m + uF p m .

On Symbol-Triple Distance of a Class of Constacyclic Codes of Length 3ps Over F pm + uF pm

“…In [19], the authors obtained the symbol-pair distances of all constacyclic codes of prime power lengths over F p m . Later, [18] and [21] gave the symbol-pair distances of all constacyclic codes of length p s over F p m + uF p m . In this VOLUME 4, 2016 section, when γ is not a cube in F p m , we determine the symbol-triple distance of all γ-constacyclic code of length 3p s over R, where the structure of γ-constacyclic codes of length 3p s over R is given in part 4 of Theorem 2.…”

On Hamming Distance Distributions of Repeated-Root Cyclic Codes of Length 5ps Over Fp m + uFp m

“…Theorem 2.8. ( [12], [19], [28]) Let C be a λ-constacyclic code of length p s over R. Then Hamming distance of all λconstacyclic codes C is determined as follows.…”

“…2) ( [19, Theorem 3.2] and [28,Appendix]) If λ ∈ F p m \ {0}, the λ-constacyclic codes of length p s over R, i.e., ideals of the ring R[x] x p s −λ have their Hamming distances completely determined as follows.…”