On the algebraic structure of quasi-cyclic codes I: Finite fields
On the algebraic structure of quasi-cyclic codes I: Finite fields
复制标题
DOI：
10.1109/18.959257
复制
发表时间：
2001-11
期刊：
影响因子：
--
通讯作者：
S. Ling;P. Solé
中科院分区：
文献类型：
--
作者：
S. Ling;P. Solé
介绍了拟循环码的一种新的代数方法。关键思想是把域上的拟循环码看作辅助环上的线性码。通过使用中国剩余定理（CRT）或离散傅立叶变换（DFT），可以将环分解为场的直接积。这种环分解反过来从较低长度的代码中产生代码结构，在某些情况下是著名的平方和立方结构，在其他情况下是（u+/spl upsi/|u-/spl upsi/）和Vandermonde结构。所有长度为3的倍数的二进制扩展二次剩余码都可以通过三次构造得到。介绍了昆亭和间隔结构。由环分解可能得到的其他结果是自对偶拟循环码的表征，以及推广循环码的迹表示。
A new algebraic approach to quasi-cyclic codes is introduced. The key idea is to regard a quasi-cyclic code over a field as a linear code over an auxiliary ring. By the use of the Chinese remainder theorem (CRT), or of the discrete Fourier transform (DFT), that ring can be decomposed into a direct product of fields. That ring decomposition in turn yields a code construction from codes of lower lengths which turns out to be in some cases the celebrated squaring and cubing constructions and in other cases the (u+/spl upsi/|u-/spl upsi/) and Vandermonde constructions. All binary extended quadratic residue codes of length a multiple of three are shown to be attainable by the cubing construction. Quinting and septing constructions are introduced. Other results made possible by the ring decomposition are a characterization of self-dual quasi-cyclic codes, and a trace representation that generalizes that of cyclic codes.