Sums of Gauss Sums and Weights of Irreducible Codes

We develop a matrix approach to compute a certain sum of Gauss sums which arises in the study of weights of irreducible codes. A lower bound on the minimum weight of certain irreducible codes is given

The Weight Distributions of a Class of Cyclic Codes with Three Nonzeros over F3

Cyclic codes have efficient encoding and decoding algorithms. The decoding error probability and the undetected error probability are usually bounded by or given from the weight distributions of the codes. Most researches are about the determination of the weight distributions of cyclic codes with few nonzeros, by using quadratic form and exponential sum but limited to low moments. In this paper, we focus on the application of higher moments of the exponential sum to determine the weight distributions of a class of ternary cyclic codes with three nonzeros, combining with not only quadratic form but also MacWilliams' identities. Another application of this paper is to emphasize the computer algebra system Magma for the investigation of the higher moments. In the end, the result is verified by one example using Matlab.Comment: 10 pages, 3 table

The weight distributions of irreducible cyclic codes of length 2m

AbstractLet m be a positive integer and q be an odd prime power. In this paper, the weight distributions of all the irreducible cyclic codes of length 2m over Fq are determined explicitly

Weight distribution of some reducible cyclic codes

AbstractLet q=pm where p is an odd prime, m⩾3, k⩾1 and gcd(k,m)=1. Let Tr be the trace mapping from Fq to Fp and ζp=e2πip. In this paper we determine the value distribution of following two kinds of exponential sums∑x∈Fqχ(αxpk+1+βx2)(α,β∈Fq) and∑x∈Fqχ(αxpk+1+βx2+γx)(α,β,γ∈Fq), where χ(x)=ζpTr(x) is the canonical additive character of Fq. As an application, we determine the weight distribution of the cyclic codes C1 and C2 over Fp with parity-check polynomial h2(x)h3(x) and h1(x)h2(x)h3(x), respectively, where h1(x), h2(x) and h3(x) are the minimal polynomials of π−1, π−2 and π−(pk+1) over Fp, respectively, for a primitive element π of Fq