Linear codes with small hulls in semi-primitive case
Linear codes with small hulls in semi-primitive case
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DOI：
10.1007/s10623-019-00663-4
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发表时间：
2019-07
期刊：
影响因子：
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通讯作者：
C. Carlet;Chengju Li;Sihem Mesnager
中科院分区：
文献类型：
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作者：
C. Carlet;Chengju Li;Sihem Mesnager
线性码的船体被定义为码与其对偶的交集，并且最初被引入用于对有限射影平面进行分类。船体在判定两个线性码的置换等价性和计算线性码的自同构群的算法复杂度方面起着重要的作用。结果表明，如果船体的尺寸较小，这些算法通常非常有效。显然，具有最小船体的线性码是LCD码，具有次小船体的线性码是具有一维船体的线性码。本文利用半本原情形下的特征和，从分圆域和有限域的乘法子群出发，构造了一维船体的LCD码和线性码。得到了这类码的一些充要条件，其中素数p在分圆域上的素理想分解起着关键作用。此外，我们证明了这些代码在某些情况下不存在。
The hull of a linear code is defined to be the intersection of the code and its dual, and was originally introduced to classify finite projective planes. The hull plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code. It has been shown that these algorithms are very effective in general if the size of the hull is small. It is clear that the linear codes with the smallest hull are LCD codes and with the second smallest hull are those with one-dimensional hull. In this paper, we employ character sums in semi-primitive case to construct LCD codes and linear codes with one-dimensional hull from cyclotomic fields and multiplicative subgroups of finite fields. Some sufficient and necessary conditions for these codes are obtained, where prime ideal decompositions of prime p in cyclotomic fields play a key role. In addition, we show the non-existence of these codes in some cases.