The Weight Distributions of Several Classes of Cyclic Codes From APN Monomials
The Weight Distributions of Several Classes of Cyclic Codes From APN Monomials
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DOI：
10.1109/tit.2014.2329694
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发表时间：
2014-06
影响因子：
2.5
通讯作者：
Chunlei Li;Nian Li;T. Helleseth;C. Ding
中科院分区：
文献类型：
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作者：
Chunlei Li;Nian Li;T. Helleseth;C. Ding
让M≥3为奇数，p是本文中的奇数。 x）是通过在参数p，m和e上检查一般条件，其中mi（x）是FP的原始元素π的最小π-I多项式。 4）和正整数满足（PK + 1）·e。 （mod pm -1）对于具有GCD（m，k）= 1的某些正整数k，指数总和t（a，b）=σx∈FpmΩtr（ax+bxe）和s（a，b）的值分布，c）=σx∈FpmΩtr（AX+BXE+CXS），其中S =（PM -1）/2作为应用程序。 Cyclic代码C（1，E，S）的重量分布，具有奇偶校验 - 检查多项式M1（X）ME（X）MS（X）。在p = 3甚至满足上述条件的情况下，环状代码C（1，e，s）的双重距离具有最佳的最小距离。
Let m ≥ 3 be an odd integer and p be an odd prime. In this paper, a number of classes of three-weight cyclic codes C(1,e) over Fp, which have parity-check polynomial m1(x)me(x), are presented by examining general conditions on the parameters p, m, and e, where mi(x) is the minimal polynomial of π-i over Fp for a primitive element π of Fpm. Furthermore, for p ≡ 3 (mod 4) and a positive integer e satisfying (pk + 1) · e ≡ 2 (mod pm - 1) for some positive integer k with gcd(m, k) = 1, the value distributions of the exponential sums T(a, b) = Σx∈Fpm ωTr(ax+bxe) and S(a, b, c) = Σx∈Fpm ωTr(ax+bxe+cxs), where s = (pm - 1)/2, are determined. As an application, the value distribution of S(a, b, c) is utilized to derive the weight distribution of the cyclic codes C(1,e,s) with parity-check polynomial m1(x)me(x)ms(x). In the case of p = 3 and even e satisfying the above condition, the dual of the cyclic code C(1,e,s) has optimal minimum distance.