1,720,964 research outputs found
On a family of linear MRD codes with parameters [8 × 8 , 16 , 7] q
Full text linkIn this paper we consider a family F of 2n-dimensional F-q-linear rank metric codes in F-q(nxn) arising from polynomials of the form x(qs) +delta x(q) (n/2 +s) is an element of F-q(n) [x]. The family F was introduced by Csajbok et al. (JAMA 548:203-220) as a potential source for maximum rank distance (MRD) codes. Indeed, they showed that F contains MRD codes for n = 8, and other subsequent partial results have been provided in the literature towards the classification of MRD codes in F for any n. In particular, the classification has been reached when n is smaller than 8, and also for n greater than 8 provided that s is small enough with respect to n. In this paper we deal with the open case n = 8, providing a classification for any large enough odd prime power q. The techniques are from algebraic geometry over finite fields, since our strategy requires the analysis of certain 3-dimensional F-q-rational algebraic varieties in a 7-dimensional projective space. We also show that the MRD codes in F are not equivalent to any other MRD codes known so far
A family of planar binomials in characteristic 2
No full textPlanar polynomials of type fa,b(x)=ax2javax.xml.bind.JAXBElement@364ca89b+1+bx2javax.xml.bind.JAXBElement@31a47e70+1, a,b∈F2javax.xml.bind.JAXBElement@6b5a89d2⁎ are investigated. In particular, all the possible pairs (a,b)∈(F2javax.xml.bind.JAXBElement@3476b478⁎)2 for which fa,b(x) is planar are determined
Bound on the order of the decomposition groups of an algebraic curve in positive characteristic
No full textIn this paper, X is an algebraic curve of genus g >= 2 defined over an algebraically closed field K of positive characteristic p, G is an automorphism group of X which fixes K element wise, and, for a point P is an element of X, G(P) is the subgroup of G which fixes P. The question "how large G(P) can be compared to g" has been the subject of several papers. We are concerned with the case where the second ramification group G(P)((2)) of G(P) is trivial. Under this condition Theorem 3.1 states that if vertical bar G(P)vertical bar > 12(g - 1) then X is either an ordinary hyperelliptic curve, or it has zero p-rank and p not equal 3. More precisely, up to birational equivalence, there exists a separable p-linearized polynomial L(T) is an element of K[T] of degree q such that an affine equation of X is L(y) = ax + 1/x with a is an element of K* in the former case, and L(y) = x(3) + bx with b is an element of K in the latter case. In 1987 Nakajima proved that if X is an ordinary curve (more generally, the second ramification group of G is trivial for every P is an element of X), then the order of G does not exceed 84g(g - 1). We show that Theorem 3.1 together with some refinements of Nakajima's computations provide a slight improvement in Nakajima's bound from 84g(g - 1) to 48(g - 1)(2). (c) 2020 Elsevier Inc. All rights reserved
A family of permutation trinomials over Fq2
No full textLet p>3 and consider a prime power q=ph. We completely characterize permutation polynomials of Fqjavax.xml.bind.JAXBElement@26b5e0b6 of the type fa,b(X)=X(1+aXq(q−1)+bX2(q−1))∈Fqjavax.xml.bind.JAXBElement@1bc6c375[X]. In particular, using connections with algebraic curves over finite fields, we show that the already known sufficient conditions are also necessary
On the weight distribution of some minimal codes
No full textMinimal codes are a class of linear codes which gained interest in the last years, thanks to their connections to secret sharing schemes. In this paper we provide the weight distributions and the parameters of families of minimal codes recently introduced by C. Tang, Y. Qiu, Q. Liao, Z. Zhou, answering some open questions
AG codes and AG quantum codes from cyclic extensions of the Suzuki and Ree curves
Full text linkWe investigate several types of linear codes constructed from two families of maximal curves over finite fields recently constructed by Skabelund as cyclic covers of the Suzuki and Ree curves. Plane models for such curves are provided, and the Weierstrass semigroup at an Fq-rational point is shown to be symmetric
Minimal linear codes from Hermitian varieties and quadrics
No full textIn this note we investigate minimal linear codes arising from Hermitian varieties and quadrics. We study their parameters and formulate some open problems about their weight distributio
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