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

机译：从有理函数域构造置换码及其对Gilbert-Varshamov界的改进

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摘要

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 of a symbol length, this class of permutation codes improves the Gilbert–Varshamov bound by a factor asymptotically for a minimum distance with . Furthermore, for a constant minimum distance , we improve the Gilbert–Varshamov bound by a factor as well as the recent one given by Gao by a factor asymptotically for all sufficiently large .
机译：由于最近在电力线，多级闪存和分组密码通信中的应用，置换码已受到编码界和数学界的广泛关注。良好排列代码的基准之一是吉尔伯特-瓦尔沙莫夫界线。尽管排列码有几种构造，但是吉尔伯特-瓦尔沙莫夫界仍然是最佳的渐近下界，除非最近在恒定最小距离的情况下有所改进。在本文中，我们提出了有理函数域中置换代码的代数构造，结果表明，对于符号长度的质数，此类置换代码将Gilbert-Varshamov边界渐近地改善了一个因子，最小与的距离。此外，对于恒定的最小距离，对于所有足够大的点，我们将Gilbert–Varshamov的渐近线提高一个因数，而对于最近由Gao给出的渐近线，则将其渐近地提高一个因数。

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