Motivated by the application of high-density data storage technologies, symbol-pair codes are proposed to protect against pair-errors in symbol-pair channels, whose outputs are overlapping pairs of symbols. The research of symbol-pair codes with the largest minimum pair-distance is interesting since such codes have the best possible error-correcting capability. A symbol-pair code attaining the maximal minimum pair-distance is called a maximum distance separable (MDS) symbol-pair code. In this paper, we focus on constructing linear MDS symbol-pair codes over the finite field F q . We show that a linear MDS symbol-pair code over F q with pair-distance 5 exists if and only if the length n ranges from 5 to q 2 + q + 1. As for codes with pair-distance 6, length ranging from 6 to q 2 +1, we construct linear MDS symbol-pair codes by using a configuration called ovoid in projective geometry. With the help of elliptic curves, we present a construction of linear MDS symbol-pair codes for any pair-distance d + 2 with length n satisfying 7 ≤ d + 2 ≤ n ≤ q + ⌊2 √ q⌋ + δ(q) − 3, where δ(q) = 0 or 1.
We determine the 2-adic complexity of the Ding-Helleseth-Martinsen (DHM) binary sequences by using cyclotomic numbers of order four, "Gauss periods" and "quadratic Gauss sum" on finite field F q and valued in Z 2 N −1 where q ≡ 5 (mod 8) is a prime number and N = 2q is the period of the DHM sequences.
Projective Reed-Solomon (PRS) codes are Reed-Solomon codes of the maximum possible length q + 1. The classification of deep holes -received words with maximum possible error distance-for PRS codes is an important and difficult problem. In this paper, we use algebraic methods to explicitly construct three classes of deep holes for PRS codes. We show that these three classes completely classify all deep holes of PRS codes with redundancy four. Previously, the deep hole classification was only known for PRS codes with redundancy at most three by the work [9].
Computing the minimum distance of a linear code is one of the fundamental problems in algorithmic coding theory. Vardy [14] showed that it is an NP-hard problem for general linear codes. In practice, one often uses codes with additional mathematical structure, such as AG codes. For AG codes of genus 0 (generalized Reed-Solomon codes), the minimum distance has a simple explicit formula. An interesting result of Cheng [3] says that the minimum distance problem is already NP-hard (under RP-reduction) for general elliptic curve codes (ECAG codes, or AG codes of genus 1). In this paper, we show that the minimum distance of ECAG codes also has a simple explicit formula if the evaluation set is suitably large (at least 2/3 of the group order). Our method is purely combinatorial and based on a new sieving technique from the first two authors [8]. This method also proves a significantly stronger version of the MDS (maximum distance separable) conjecture for ECAG codes.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.