A Construction of Permutation Codes From Rational Function Fields and Improvement to the Gilbert–Varshamov Bound

Abstract

Due to recent applications to communications over powerlines, multilevel flash memories, and block ciphers, permutation codes have received a lot of attention from both coding and mathematical communities. One of the benchmarks for good permutation codes is the Gilbert–Varshamov bound. Although there have been several constructions of permutation codes, the Gilbert–Varshamov bound still remains to be the best asymptotical lower bound except for a recent improvement in the case of constant minimum distance. In this paper, we present an algebraic construction of permutation codes from rational function fields, and it turns out that, for a prime number $n$ of a symbol length, this class of permutation codes improves the Gilbert–Varshamov bound by a factor $n$ asymptotically for a minimum distance $d$ with $d=O(\sqrt {n})$ . Furthermore, for a constant minimum distance $d$ , we improve the Gilbert–Varshamov bound by a factor $n$ as well as the recent one given by Gao et al. by a factor $n/\log n$ asymptotically for all sufficiently large $n$ .