New Constructions of MDS Codes With Complementary Duals

Abstract

Linear complementary-dual (LCD for short) codes are linear codes that intersect with their duals trivially. LCD codes have been used in certain communication systems. It is recently found that LCD codes can be applied in cryptography. This application of LCD codes renewed the interest in the construction of LCD codes having a large minimum distance. MDS codes are optimal in the sense that the minimum distance cannot be improved for given length and code size. Constructing LCD MDS codes is thus of significance in theory and practice. Recently, Jin ([8], IEEE Trans. Inf. Theory, 2016) constructed several classes of LCD MDS codes through generalized Reed-Solomon codes. In this paper, a different approach is proposed to obtain new LCD MDS codes from generalized Reed-Solomon codes. Consequently, new code constructions are provided and certain previously known results in [8] are extended.

MDS Codes With Hulls of Arbitrary Dimensions and Their Quantum Error Correction

“…In this section, utilizing generalized Reed-Solomon codes, we provide several constructions of MDS codes with arbitrary dimensional hull. As pointed out in the first section, the existence question about MDS codes with complementary duals over F q has been completely addressed in [4] and [11]. Here we only consider the case that the dimension of hull is greater than 0.…”

“…To the best of our knowledge, it is the first infinite family of MDS EAQECCs that the required number of maximally entangled states can take all values since dim(Hull(C)) is arbitrary. (we exclude the case that dim(Hull(C)) = 0, i.e., c = k or n − k since the existence question about MDS codes with 0-dimensional hull has been completely solved for q = 2 and q > 3 in [4] and [11]. )…”

“…Recently, the study of LCD codes has attracted much attention due to their applications in orthogonal direct sum masking (ODSM), protecting against side-channel attacks (SCAs) and fault injection attacks (FIAs). The existence question about MDS codes with complementary duals over F q has been completely addressed in [4] and [11] for q = 2 and q > 3, respectively. The test of the permutation equivalence of two codes and the determination of the automorphism group of a linear code are interesting problems in coding theory [14,17].…”

“…The purpose of this paper is to construct linear codes from generalized Reed-Solomon (GRS) codes or extended generalized Reed-Solomon codes and determine their hull. Inspired by the idea of [5], we propose several constructions of MDS codes with arbitrary dimensional hull. Furthermore, by using these MDS codes with arbitrary dimensional hull, we obtain several new infinite families of MDS EAQECCs.…”

Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs

“…A practical application of binary LCDs against sidechannel attacks (SCAs) and fault injection attacks (FIAs) was investigated by Carlet et al in [10] and [11]. The study of LCD codes is thus becoming a hot research topic in coding theory ( [12,13,14,15,16]). A surprising result was given in [16], which proved that any linear code over F q (q > 3) is equivalent to a Euclidean LCD code and any linear code over F q 2 (q > 2) is equivalent to a Hermitian LCD code.…”