Applicable Algebra in Engineering Communication and Computing最新文献

Abstract

As an important class of cyclic codes, BCH codes are widely employed in satellite communications, DVDs, CD, DAT etc. In this paper, we determine the dimension of BCH codes of length (frac{q^{2m}-1}{q+1}) over the finite fields ({mathbb {F}}_q) . We settle a conjecture about the largest q-cyclotomic coset leader modulo n which was proposed by Wu et al. We also get the second largest q-cyclotomic coset leader modulo n if m is odd. Moreover, we investigate the parameters of ({mathcal {C}}_{(n,q,delta _i)}) ( ({mathcal {C}}_{(n,q,delta _i)}^perp) ) for (i=1,2) and ({mathcal {C}}_{(n,q,delta )}) ( ({mathcal {C}}_{(n,q,delta )}^perp) ) for (2le delta le q-1) . What’s more, we obtain many (almost) optimal codes.

Linear codes are widely studied due to their important applications in authentication codes, association schemes and strongly regular graphs, etc. In this paper, a class of at most three-weight linear codes is constructed by selecting a new defining set, then the parameters and weight distributions of codes are determined by exponential sums. Results show that almost all the linear codes we constructed are minimal and we also describe the access structures of the secret sharing schemes based on their dual. Especially, the new binary code is a two-weight projective code and based on which a strongly regular graph with new parameters is designed.

As a class of linear codes, cyclic codes are widely used in communication systems, consumer electronics and data storage systems due to their favorable properties. In this paper, we construct two classes of optimal p-ary cyclic codes with parameters ([p^m-1, p^m-frac{3m}{2}-2, 4]) by analyzing the solutions of certain polynomials over finite fields. Furthermore, we propose an efficient method to determine the optimality of the 7-ary cyclic code (mathcal {C}_{(0,1,e)}) and present three classes of optimal codes with parameters ([7^m-1,7^m-2m-2,4]). Additionally, we provide the weight distribution of one class of their duals.

At present, GLV/GLS scalar multiplication mainly focuses on elliptic curves in Weierstrass form, attempting to find and construct more and more efficiently computable endomorphism. In this paper, we investigate the application of the GLV/GLS scalar multiplication technique to Jacobi Quartic curves. Firstly, we present a concrete construction of efficiently computable endomorphisms for this type of curves over prime fields by exploiting birational equivalence between curves, and obtain a 2-dimensional GLV method. Secondly, we consider the quadratic twists of elliptic curves. By using birational equivalence and Frobenius mapping between curves, we present methods to construct efficiently computable endomorphisms for this type of curves over the quadratic extension field, and obtain a 2-dimensional GLS method. Finally, we obtain a 4-dimensional GLV method on elliptic curves with j-invariant 0 or 1728 by using higher degree twists. The experimental results show that the speedups of the 2-dimensional GLV method and 4-dimensional GLV method compared to 5-NAF method exceed (37.2%) and (109.4%) for Jacobi Quartic curves, respectively. At the same time, when utilizing one of the proposed methods, the scalar multiplication on Jacobi Quartic curves is consistently more efficient than on elliptic curves in Weierstrass form.