Applicable Algebra in Engineering Communication and Computing最新文献

This paper continues previous ones about compression of points on elliptic curves (E_b!: y^2 = x^3 + b) (with j-invariant 0) over a finite field (mathbb {F}_{!q}) of characteristic (p > 3). It is shown in detail how any two (resp., three) points from (E_b(mathbb {F}_{!q})) can be quickly compressed to two (resp., three) elements of (mathbb {F}_{!q}) (apart from a few auxiliary bits) in such a way that the corresponding decompression stage requires to extract only one cubic (resp., sextic) root in (mathbb {F}_{!q}). As a result, for many fields (mathbb {F}_{!q}) occurring in practice, the new compression-decompression methods are more efficient than the classical one with the two (resp., three) x or y coordinates of the points, which extracts two (resp., three) roots in (mathbb {F}_{!q}). As a by-product, it is also explained how to sample uniformly at random two (resp., three) “independent” (mathbb {F}_{!q})-points on (E_b) essentially at the cost of only one cubic (resp., sextic) root in (mathbb {F}_{!q}). Finally, the cases of four and more points from (E_b(mathbb {F}_{!q})) are commented on as well.

A piecewise linear function can be described in different forms: as a nested expression of (min)- and (max)-functions, as a difference of two convex piecewise linear functions, or as a linear combination of maxima of affine-linear functions. In this paper, we provide two main results: first, we show that for every piecewise linear function (f:{mathbb{R}}^{n} rightarrow {mathbb{R}}), there exists a linear combination of (max)-functions with at most (n+1) arguments, and give an algorithm for its computation. Moreover, these arguments are contained in the finite set of affine-linear functions that coincide with the given function in some open set. Second, we prove that the piecewise linear function (max (0, x_{1}, ldots , x_{n})) cannot be represented as a linear combination of maxima of less than (n+1) affine-linear arguments. This was conjectured by Wang and Sun (IEEE Trans Inf Theory 51:4425–4431, 2005) in a paper on representations of piecewise linear functions as linear combination of maxima.

Many sequences that arise in combinatorics and the analysis of algorithms turn out to be holonomic (note that some authors prefer the terminology D-finite). In this paper, we study various basic algorithmic problems for such sequences ((f_n)_{n in {mathbb {N}}}): how to compute their asymptotics for large n? How to evaluate (f_n) efficiently for large n and/or large precisions p? How to decide whether (f_n > 0) for all n? We restrict our study to the case when the generating function (f = sum _{n in {mathbb {N}}} f_n z^n) satisfies a Fuchsian differential equation (often it suffices that the dominant singularities of f be Fuchsian). Even in this special case, some of the above questions are related to long-standing problems in number theory. We will present algorithms that work in many cases and we carefully analyze what kind of oracles or conjectures are needed to tackle the more difficult cases.

Quaternary sequences with high 4-adic complexity have attracted widespread attention in communication and cryptography systems. In this paper, for an odd prime p,  several new classes of quaternary sequences with even period 2p are constructed by using the Legendre sequences of period p. And the autocorrelation distribution of the new quaternary sequences is derived. We determine the 4-adic complexity of these new quaternary sequences. The results show that the new quaternary sequences have good autocorrelation properties, and the 4-adic complexity of these new quaternary sequences is large enough to resist the attack of the rational approximation algorithm.

Given a finite group G with the identity e, the proper intersection power graph of G is the graph with vertex set (Gsetminus {e}), in which two distinct vertices x and y are adjacent if (langle xrangle cap langle yrangle ) is non-trivial. In this paper, we give a necessary and sufficient condition for a proper intersection power graph to contain a perfect code. As applications, we classify all finite nilpotent groups whose proper intersection power graphs admit a perfect code. We also classify a few classes of finite non-nilpotent groups whose proper intersection power graphs admit a perfect code, such as, symmetric groups and alternating groups.

Array-designed codes are capable of correcting row and column deletions. In this paper, we introduce array-designed reversible and complementary codes over the finite field GF(4), which can correct row deletions and column deletions errors. Hence our codes can be useful for DNA codes. We also propose a construction method of array-designed reversible and complementary codes. As examples, we describe the tensor product of cyclic LCD codes and the tensor product of self-dual reversible codes over GF(4).

This note is just a modest contribution to prove several classical results in Combinatorics from notions of Duality in some Artinian K-algebras (mainly through the Trace Formula), where K is a perfect field of characteristics not equal to 2. We prove how several classic combinatorial results are particular instances of a Trace (Inversion) Formula in finite (mathbb {Q})-algebras. This is the case with the Exclusion-Inclusion Principle (in its general form, both with direct and reverse order associated to subsets inclusion). This approach also allows us to exhibit a basis of the space of null t-designs, which differs from the one described in Theorem 4 of Deza and Frankl (Combinatorica 2:341–345, 1982). Provoked by the elegant proof (which uses no induction) in Frankl and Pach (Eur J Comb 4:21–23, 1983) of the Sauer–Shelah–Perles Lemma, we produce a new one based only in duality in the (mathbb {Q})-algebra (mathbb {Q}[V_n]) of polynomials functions defined on the zero-dimensional algebraic variety of subsets of the set ([n]:={1,2,ldots , n}). All results are equally true if we replace (mathbb {Q}[V_n]) by (K[V_n]), where K is any perfect field of characteristics (not =2). The article connects results from two fields of mathematical knowledge that are not usually connected, at least not in this form. Thus, we decided to write the manuscript in a self-contained survey-like style, although it is not a survey paper at all. Readers familiar with Commutative Algebra probably know most of the proofs of the statements described in section 2. We decided to include these proofs for those potential readers not so familiar with this framework.

Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an important subject of study for many years. Recently, several classes of optimal quinary cyclic codes of the forms (mathcal {C}_{(0,1,e)}) and (mathcal {C}_{(1,e,s)}) are presented in the literature, where (s=frac{5^m-1}{2}) and (2 le e le 5^{m}-2). In this paper, by considering the solutions of certain equations over finite fields, we give three new classes of infinite families of optimal quinary cyclic codes of the form (mathcal {C}_{(1,e,s)}) with parameters ([5^{m}-1, 5^{m}-2m-2, 4]). Specifically, we make progress towards an open problem proposed by Gaofei Wu et al. [17].

A relationship between a parametric ideal and its generic Gröbner basis is discussed, and a new stability condition of a Gröbner basis is given. It is shown that the ideal quotient of the parametric ideal by the ideal generated using the generic Gröbner basis provides the stability condition. Further, the stability condition is utilized to compute a comprehensive Gröbner system, and hence as an application, we obtain a new algorithm for computing comprehensive Gröbner systems. The proposed algorithm has been implemented in the computer algebra system Risa/Asir and experimented with a number of examples. Its performance (execution timings) has been compared with the Kapur-Sun-Wang’s algorithm for computing comprehensive Gröbner systems.