Weight Distribution of Cosets of Small Codes With Good Dual Properties
Weight Distribution of Cosets of Small Codes With Good Dual Properties
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DOI：
10.1109/tit.2015.2487348
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发表时间：
2014-08
影响因子：
2.5
通讯作者：
L. Bazzi
中科院分区：
文献类型：
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作者：
L. Bazzi
二进制线性码的双边最小距离是使得所有非零码字具有d和n-d之间的权重的最大值d。设Q ∈ {0，1}n是二元线性码，其对偶码的双边最小距离至少为d，其中d为奇数。粗略地说，我们证明了Q的随机陪集的权分布与二项分布之间的平均L∞-距离，即L1-距离，随着Q的对偶的双边最小距离d的增加而迅速衰减.对于d = n（1），它像n-n（d）一样衰减。在另一个d = n（n）的极端，它像和e-n（d）一样衰减。因此，几乎所有的陪集Q的权重分布非常接近二项分布。特别是，我们建立了以下界限。若Q的对偶有至少d = 2 t + 1的双边最小距离，其中t ≥ 1为整数，则平均L∞-距离至多为min{（e ln（n/2 t））t（2 t/n）（t/2），<$2e-（t/10）}.对于平均L1-距离，我们得到了min{（2 t + 1）（e ln（n/2 t））t（2 t/n）（t/2）-1，<$2（n + 1）e-（t/10）}的界，当t ≥ 3时，它给出了非平凡的结果.我们给出了扩展Hadamard码和扩展对偶BCH码陪集的重量分布的应用。我们的论点是基于傅立叶分析，线性规划和多项式逼近技术。
The bilateral minimum distance of a binary linear code is the maximum d such that all nonzero codewords have weights between d and n - d. Let Q ⊂ {0,1}n be a binary linear code whose dual has bilateral minimum distance at least d, where d is odd. Roughly speaking, we show that the average L∞-distance-and consequently, the L1-distance-between the weight distribution of a random cosets of Q and the binomial distribution decays quickly as the bilateral minimum distance d of the dual of Q increases. For d = ⊖(1), it decays like n-⊖(d). On the other d = ⊖(n) extreme, it decays like and e-⊖(d). It follows that, almost all cosets of Q have weight distributions very close to the to the binomial distribution. In particular, we establish the following bounds. If the dual of Q has bilateral minimum distance at least d = 2t + 1, where t ≥ 1 is an integer, then the average L∞-distance is at most min{(e ln (n/2t))t(2t/n)(t/2), √2e-(t/10)}. For the average L1-distance, we conclude the bound min{(2t + 1)(e ln (n/2t))t(2t/n)(t/2)-1, √2(n + 1)e-(t/10)}, which gives nontrivial results for t ≥ 3. We give applications to the weight distribution of cosets of extended Hadamard codes and extended dual BCH codes. Our argument is based on Fourier analysis, linear programming, and polynomial approximation techniques.