On Determining Deep Holes of Generalized Reed-Solomon Codes

Abstract

Abstract. For a linear code, deep holes are defined to be vectors that are further away from codewords than all other vectors. The problem of deciding whether a received word is a deep hole for generalized ReedSolomon codes is proved to be co-NP-complete [9] [5]. For the extended Reed-Solomon codes RSq(Fq, k), a conjecture was made to classify deep holes in [5]. Since then a lot of effort has been made to prove the conjecture, or its various forms. In this paper, we classify deep holes completely for generalized Reed-Solomon codes RSp(D, k), where p is a prime, |D| > k p−1 2. Our techniques are built on the idea of deep hole trees, and several results concerning the Erdös-Heilbronn conjecture.

Deep holes in Reed–Solomon codes based on Dickson polynomials

“…Cheng, Li, and Zhuang in the previously referenced [4] were able to resolve this under some conditions. Theorem 6.…”

“…In one of the newest papers, Cheng, Li, and Zhuang [4] were able to resolve in some cases the conjecture over F p using the concept of deep hole trees.…”

On Deep Holes of Projective Reed-Solomon Codes over Finite Fields with Even Characteristic

“…Lemma 4 [16] Let C be [nk] maximal distance separable code over finite field F q . If the covering radius ρ(C) of code C is nk, then the received vector u Î F n q is a deep hole of C if and only if any k + 1 columns of the (k + 1)´n matrix ( ) G u is linearly independent, where G is the generator matrix of code C.…”

On the error distance of extended Reed-Solomon codes

“…So far, no one has proved or defied Cheng-Murray's conjecture on extended Reed-Solomon code. Recently, Cheng, Li and Zhuang [3] classified deep holes completely for generalized Reed-Solomon codes C q (D, k), where q is a prime, and |D| > k ≥ q−1 2 . Moreover, they proved the the following result.…”