On The Weights of Irreducible Cyclic Codes
On The Weights of Irreducible Cyclic Codes
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发表时间：
2005
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通讯作者：
Y. Aubry
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作者：
Y. Aubry
本文主要研究不可约循环码的重量分布。我们从解释由于McEliece这些权重的高斯总和。首先，利用Stickelberger同余和Gross-Koblitz公式进行p-adic分析，通过给出权重的结果改进了McEliece整除定理.其次，结合施密特和白色的一个猜想，我们研究了指数为2的二元不可约循环码。我们表明，假设广义黎曼假设，有无限多个这样的代码。此外，我们考虑这个家庭的代码，称为二次剩余码的子类。这些码的参数与某些虚二次数域的类数有关。给出了指数为2的二元不可约循环码不存在的一个初等证明。
The paper is devoted to the study of the weight distribution of irreducible cyclic codes. We start from the interpretation due to McEliece of these weights by means of Gauss sums. Firstly, a p-adic analysis using Stickelberger congruences and Gross-Koblitz formula enable us to improve the divisibility theorem of McEliece by giving results on the multiplicity of the weights. Secondly, in connection with a conjecture of Schmidt and White, we focus on index 2 binary irreducible cyclic codes. We show, assuming the generalized Riemann hypothesis, that there are infinitely many such codes. Furthermore, we consider a subclass of this family of codes, called Quadratic Residue codes. The parameters of these codes are related to class numbers of some imaginary quadratic number fields. We give an elementary proof of the non-existence of two-weight binary irreducible cyclic codes of index 2.