Deep holes of generalized Reed-Solomon codes

Abstract

In this paper, deep holes of Reed-Solomon (RS) codes are studied. A new class of deep holes for generalized affine RS codes is given if the evaluation set satisfies certain combinatorial structure. Three classes of deep holes for projective Reed-Solomon (PRS) codes are constructed explicitly. In particular, deep holes of PRS codes with redundancy three are completely obtained when the characteristic of the finite field is odd. Most (asymptotically of ratio 1) of the deep holes of PRS codes with redundancy four are also obtained.

On the error distance of extended Reed-Solomon codes

“…Zhang, Fu and Liao [15] extended the result above to any evaluation set D = F q , and derived the following conclusion.…”

Deep Holes of Projective Reed-Solomon Codes

“…Deciding deep holes of a given code is much harder than the covering radius problem, even for RS codes. The deep hole problem for RS codes was studied in [4], [5], [14], [15], [17], [18], [24], [25], [26]. As noted above, words u f with deg(f ) = k are deep hole of RS(D, k).…”

Rational points on complete symmetric hypersurfaces over finite fields

“…is quadratic. It was shown in [13,Theorem 4.2] that N * q (D f ) > 0 for 3 ≤ k ≤ q − 2 (k = q − 2 if q is even). Conjecture 2.2 is equivalent to the following conjecture.…”