On the error distance of extended Reed-Solomon codes

Abstract

It is well known that the main problem of decoding the extended Reed-Solomon codes is computing the error distance of a word. Using some algebraic constructions, we are able to determine the error distance of words whose degrees are k + 1 and k + 2 to the extended Reed-Solomon codes. As a corollary, we can simply get the results of Zhang-Fu-Liao on the deep hole problem of Reed-Solomon codes.

On Deep Holes of Elliptic Curve Codes

“…It is much harder than the covering radius problem, even for affine RS codes. The deep hole problem for Reed-Solomon codes was studied in [4,6,16,17,18,20,21,22,31,32,33,34]. For Reed-Solomon codes of length n much smaller than q, deciding if a given word is a deep hole is equivalent to a general subset sum problem, which is NP-hard.…”

Counting polynomials with distinct zeros in finite fields

“…As an application, we define two kinds of Wenger graphs called jumped Wenger graphs and obtain their explicit spectrum. Note that in the cases n − ℓ = 2, 3, the smallest k such that N k (u(x), ℓ) > 0 is determined in [15] by giving an explicit construction of a solution. Our result gives an exact and explicit formula for N k (u(x), ℓ).…”