Deep holes in Reed–Solomon codes based on Dickson polynomials

Abstract

For an [n, k] Reed-Solomon code C, it can be shown that any received word r lies a distance at most n − k from C, denoted d(r, C) ≤ n − k. Any word r meeting the equality is called a deep hole. Guruswami and Vardy (2005) showed that for a specific class of codes, determining whether or not a word is a deep hole is NP-hard. They suggested passingly that it may be easier when the evaluation set of C is large or structured. Following this idea, we study the case where the evaluation set is the image of a Dickson polynomial, whose values appear with a special uniformity. To find families of received words that are not deep holes, we reduce to a subset sum problem (or equivalently, a Dickson polynomial-variation of Waring's problem) and find solution conditions by applying an argument using estimates on character sums indexed over the evaluation set.

On 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).…”

Moment subset sums over finite fields

“…Following established procedures, we use the Li-Wan sieve [17] to analyze large values of k. This method has been used several times [26,14,17,18,19,24] and is now standard. So, we will only give an outline and indicate the differences.…”

“…The following lemma is the key character sum estimate we need. The proof follows the method used in [14], where the case m = 1 is treated.…”

On deep holes of Gabidulin codes

“…The covering radius and deep holes of a linear code embedded with Hamming metric were studied extensively [1,2,3,4,5,10,12,14,16,22,23,24,25,26,27], in which MDS codes such as generalized Reed-Solomon codes, standard Reed-Solomon codes and projective Reed-Solomon codes were explored deeply. Gabidulin codes were introduced by Gabidulin in [7] and independently by Delsarte in [6].…”