Distance Distribution in Reed-Solomon Codes
Jiyou Li, Daqing Wan · IEEE Transactions on Information Theory · 2019
Let Fq be the finite field of q elements. In this paper we obtain bounds on the following counting problem: given a polynomial f (x) ∈ Fq[x] of degree k + m and a non-negative integer r, count the number of polynomials g(x) ∈ Fq[x] of degree at most k - 1 such that f (x) + g(x) has exactly r roots in Fq. Previously, explicit formulas were known only for the cases m = 0, 1, 2. As an application, we obtain an asymptotic formula on the list size of the standard Reed-Solomon code [q, k, q - k + 1]q.