Complementary Dual Algebraic Geometry Codes
TL;DR: In this paper, a construction scheme for obtaining linear complementary dual (LCD) codes from any algebraic curve is presented, and some explicit LCD codes from elliptic curves are presented.
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Abstract: Linear complementary dual (LCD) codes are a class of linear codes introduced by Massey in 1964. LCD codes have been extensively studied in literature recently. In addition to their applications in data storage, communications systems, and consumer electronics, LCD codes have been employed in cryptography. More specifically, it has been shown that LCD codes can also help improve the security of the information processed by sensitive devices, especially against so-called side-channel attacks (SCA) and fault non-invasive attacks. In this paper, we are interested in the construction of particular algebraic geometry LCD codes which could be good candidates to be resistant against SCA. We firstly provide a construction scheme for obtaining LCD codes from any algebraic curve. Then, some explicit LCD codes from elliptic curves are presented. Maximum distance separable (MDS) codes are of the most importance in coding theory due to their theoretical significance and practical interests. In this paper, all the constructed LCD codes from elliptic curves are MDS or almost MDS. Some infinite classes of LCD codes from elliptic curves are optimal due to the Griesmer bound. Finally, we also derive some explicit LCD codes from hyperelliptic curves and Hermitian curves.
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Citations
Linear Codes Over $\mathbb F_q$ Are Equivalent to LCD Codes for $q>3$
TL;DR: In this paper, it was shown that linear linear codes with complementary duals (LCD) are equivalent to Hermitian linear codes and Euclidean linear codes (e.g., linear linear code with complementary non-dual duals).
224
Construction of MDS Codes With Complementary Duals
TL;DR: The main purpose of this paper is to construct several classes of M DS codes with complimentary duals, i.e., LCD MDS codes, through generalized Reed-Solomon codes.
122
Euclidean and Hermitian LCD MDS codes
TL;DR: In this paper, the existence of q-ary [n, k] LCD MDS codes for various lengths n and dimensions k was studied and solved for the Euclidean case.
113
New Constructions of MDS Codes With Complementary Duals
Bocong Chen,Hongwei Liu +1 more
TL;DR: A different approach is proposed to obtain new LCD MDS codes from generalized Reed-Solomon codes, and new code constructions are provided and certain previously known results by Jin are extended.
104
New Characterization and Parametrization of LCD Codes
TL;DR: In this article, a new characterization of binary linear complementary dual (LCD) cyclic codes in terms of their orthogonal or symplectic basis is presented, and a conjecture proposed by Galvez et al. on the minimum distance of binary LCD codes is solved.
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Linear codes with complementary duals
TL;DR: The nearest-neighbor decoding problem for LCD codes is shown to reduce to the problem: given a word in C⊥, find the nearest codeword in C.
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Counting points on elliptic curves over finite fields
TL;DR: Three algorithms to count the number of points on an elliptic curve over a nite eld are described, based on Shanks's baby-step-giant-step strategy, the endomorphism ring of the curve is known and several practical improvements by Atkin and Elkies are discussed.
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