A Family of Five-Weight Cyclic Codes and Their Weight Enumerators

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, a family of $p$-ary cyclic codes whose duals have three zeros are proposed. The weight distribution of this family of cyclic codes is determined. It turns out that the proposed cyclic codes have five nonzero weights.Comment: 14 Page

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