It was shown by Massey that linear complementary dual (LCD for short) codes are asymptotically good. In 2004, Sendrier proved that LCD codes meet the asymptotic Gilbert-Varshamov (GV for short) bound. Until now, the GV bound still remains to be the best asymptotical lower bound for LCD codes. In this paper, we show that an algebraic geometry code over a finite field of even characteristic is eq...

A very popular trend in code-based cryptography is to decrease the public-key size by focusing on subclasses of alternant/Goppa codes which admit a very compact public matrix, typically quasi-cyclic (QC), quasi-dyadic (QD), or quasi-monoidic (QM) matrices. We show that the very same reason which allows to construct a compact public-key makes the key-recovery problem intrinsically much easier. T...

The stabilizer code is the most general algebraic construction of quantum error-correcting codes proposed so far. A stabilizer code can be constructed from a self-orthogonal subspace of a symplectic space over a finite field. We propose a construction method of such a selforthogonal space using an algebraic curve. By using the proposed method we construct an asymptotically good sequence of bina...

Using concepts of physics of elementary particles concerning the breaking of symmetry and grannd unified theory we propose to study with the algebraic approximation the degeneracy finded in the genetic code with the incorporation of a horizontal symmetry used in gauge theories to fit the contents of the multiplets of the genetic code. It is used the algebraic approch of Hornos et. al. [1, 2, 3,...

A new method for fast decoding when using algebraic codes for the fixed codebook in CELP speech coders is presented. This method is based on the trellis structure of a block code, and allows fast optimal search of the residual codebook even with a combined scalar gain, unlike previous search methods. The method is flexible, allowing for long block lengths and the use of any code including nonli...

An (n, m, k)-resilient function is a function f : F2 → F2 such that every possible output m-tuple is equally likely to occur when the values of k arbitrary inputs are fixed by an adversary and the remaining n − k input bits are chosen independently at random. In this paper we propose a new method to generate a (n + D + 1, m, d − 1)-resilient function for any non-negative integer D whenever a [n...

Any linear code with a nontrivial automorphism has the structure of a module over a polynomial ring. The theory of Griihner bases for modules gives a compact description and implementation of a systematic encoder. We present examples of algebraic-geometric Goppa codes that can be encoded by these methods, including the one-point Hermitian codes. Index TermsSystematic encoding, algebraic-geometr...

In the early eighties tools from algebraic geometry were applied by V. Goppa to construct linear codes using algebraic curves over finite fields, see [7]. Nowadays these codes are called algebraic-geometric codes, AG codes for short. The starting point in the construction of an AG code is a projective, absolutely irreducible, non singular algebraic curve X of genus g ≥ 1 defined over the finite...

We have seen in [13] a new reduction method for the decoding of 2×2 algebraic space-time codes, called algebraic reduction as been introduced. Algebraic codes such that the volume of the Dirichlet’s polyhedron of its units group is small are better suited for decoding using the method of algebraic reduction. In this paper, we propose a new framework for constructing a space-time code whose alge...