All Two-Weight Irreducible Cyclic Codes?
All Two-Weight Irreducible Cyclic Codes?
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DOI：
10.1006/ffta.2000.0293
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发表时间：
2002
影响因子：
1
通讯作者：
Bernhard Schmidt;Clinton L White
中科院分区：
文献类型：
--
作者：
Bernhard Schmidt;Clinton L White
本文的目的是对双权不可约循环码进行分类。利用傅立叶变换和高斯和，得到了不可约循环码至多有两个权的充要数值条件。这给出了一个统一的解释，所有的两个重量不可约循环码，并允许一个完全的分类。除了两个已知的无限家庭的两个重量不可约循环码，计算机搜索显示11零星的例子。我们猜想这些码都是双权不可约循环码，并在GRH上给出了猜想的部分条件证明。
The aim of this paper is the classification of two-weight irreducible cyclic codes. Using Fourier transforms and Gauss sums, we obtain necessary and sufficient numerical conditions for an irreducible cyclic code to have at most two weights. This gives a unified explanation for all two-weight irreducible cyclic codes and allows a conjecturally complete classification. Aside from the two known infinite families of two-weight irreducible cyclic codes, a computer search reveals 11 sporadic examples. We conjecture that these are already all two-weight irreducible cyclic codes and give a partial proof of our conjecture conditionally on GRH.