Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual are trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. In this paper, we introduce a general construction of LCD codes from any linear codes. Further, we show that any linear code over Fq(q > 3) is equivalent to an Euclidean LCD code and any linear code over F q 2 (q > 2) is equivalent to a Hermitian LCD code. Consequently an [n, k, d]-linear Euclidean LCD code over Fq with q > 3 exists if there is an [n, k, d]-linear code over Fq and an [n, k, d]-linear Hermitian LCD code over F q 2 with q > 2 exists if there is an [n, k, d]-linear code over F q 2 . Hence, when q > 3 (resp.q > 2) q-ary Euclidean (resp. q 2 -ary Hermitian) LCD codes possess the same asymptotical bound as q-ary linear codes (resp. q 2 -ary linear codes). Finally, we present an approach of constructing LCD codes by extending linear codes.

The concept of an error-correcting array gives a new bound on the minimum distance of linear codes and a decoding algorithm which decodes up to half this bound. This gives a unified point of view which explains several improvements on the minimum distance of algebraic-geometric codes. Moreover it is explained in terms of linear algebra and the theory of semigroups only.

The order bound on generalized Hamming weights is introduced in a general setting of codes on varieties which comprises both the one point geometric Goppa codes as the q-ary Reed-Muller codes. For the latter codes it is shown that this bound is sharp and that they satisfy the double chain condition.

This paper defines the q-analogue of a matroid and establishes several properties like duality, restriction and contraction. We discuss possible ways to define a q-matroid, and why they are (not) cryptomorphic. Also, we explain the motivation for studying q-matroids by showing that a rank metric code gives a q-matroid. This paper establishes the definition and several basic properties of q-matroids. Also, we explain the motivation for studying q-matroids by showing that a rank metric code gives a q-matroid. We give definitions of a q-matroid in terms of its rank function and independent spaces. The dual, restriction and contraction of a q-matroid are defined, as well as truncation, closure, and circuits. Several definitions and results are straightforward translations of facts for ordinary matroids, but some notions are more subtle. We illustrate the theory by some running examples and conclude with a discussion on further research directions involving q-matroids.Many theorems in this article have a proof that is a straightforward qanalogue of the proof for the case of ordinary matroids. Although this makes them appear very easy, we feel it is needed to include them for completeness and also because it is not a guarantee that q-analogues of proofs exist.

The notion of an order domain is generalized. The behaviour of an order domain by taking a subalgebra, the extension of scalars, and the tensor product is studied. The relation of an order domain with valuation theory, Gr .o obner algebras, and graded structures is given. The theory of Gr .o obner bases for order domains is developed and used to show that the factor ring theorem and its converse, the presentation theorem, hold. The dimension of an order domain is related to the rank of its value semigroup. # 2002 Elsevier Science (USA)

This paper investigates the generalized rank weights, with a definition implied by the study of the generalized rank weight enumerator. We study rank metric codes over $L$, where $L$ is a finite Galois extension of a field $K$. This is a generalization of the case where $K = \mathbb{F}_q$ and $L = \mathbb{F}_{q^m}$ of Gabidulin codes to arbitrary characteristic. We show equivalence to previous definitions, in particular the ones by Kurihara-Matsumoto-Uyematsu, Oggier-Sboui and Ducoat. As an application of the notion of generalized rank weights, we discuss codes that are degenerate with respect to the rank metric.Comment: 15 pages; extended abstract accepted for presentation at ACA2015 (http://www.usthb.dz/spip.php?article1039

The q-ary Reed-Muller (RM) codes RM (u; m) of length n = q are a generalization of Reed-Solomon (RS) codes, which use polynomials in m variables to encode messages through functional encoding. Using an idea of reducing the multivariate case to the univariate case, randomized list-decoding algorithms for RM codes were given in [1] and [15]. The algorithm in [15] is an improvement of the algorithm in [1], it is applicable to codes RM (u; m) with u < q=2 and works for up to E < n(1 0 2u=q) errors. In this correspondence, following [6], we show that q-ary RM codes are subfield subcodes of RS codes over. Then, using the list-decoding algorithm in [5] for RS codes over , we present a list-decoding algorithm for q-ary RM codes. This algorithm is applicable to codes of any rates, and achieves an error-correction bound n(1 0 (n 0 d)=n). The algorithm achieves a better error-correction bound than the algorithm in [15], since when u is small

Abstract. We present a variant of the Diffie-Hellman scheme in which the number of bits exchanged is one third of what is used in the classical Diffie-Hellman scheme, while the offered security against attacks known today is the same. We also give applications for this variant and conjecture a extension of this variant further reducing the size of sent information.