Applicable Algebra in Engineering Communication and Computing最新文献

An ideal ({{mathcal {I}}}) of ({mathbb {Q}}[x_1,dots ,x_n]) is said to be m-absorbing if any monomial of total degree (p>m) belongs to ({{mathcal {I}}}) and if there is a monomial M of total degree m such that (Mnot in {{mathcal {I}}}.) Inspired by the fundamental work of Lechuga and Murillo (Topology 39:89–94, 2000) who established a connection between graph theory and rational homotopy theory, these paper aims to characterise the m-absorbing ideals of ({mathbb {Q}}[x_1,dots ,x_n]) generated by a family of complete homogeneous symmetric polynomials of the form (P^{(k)}_{rs}=sum _{t=1}^{k}x^{k-t}_{r}x^{t-1}_{s}), where (kge 3.)

We revisit the seminal Brill–Noether algorithm for plane curves with ordinary singularities. Our new approach takes advantage of fast algorithms for polynomials and structured matrices. We design a new probabilistic algorithm of Las Vegas type that computes a Riemann–Roch space in expected sub-quadratic time.

Moved by a question posed us by Wolfgang Rump, we investigate the Rump ideal ({mathbb {I}}(p^2-pq+qp)subset {mathbb {Z}}langle q,q^{-1}, prangle ) and we show, this way, the power of Zacharias representation.

Computer Algebra relies heavily on the computation of Gröbner bases, and these computations are primarily performed by means of Buchberger’s algorithm. In this overview paper, we focus on methods avoiding the computational intensity associated to Buchberger’s algorithm and, in most cases, even avoiding the concept of Gröbner bases, in favour of methods relying on linear algebra and combinatorics.

Let r be a divisor of (q-1.) An element (alpha in {mathbb {F}}_{q}) is said to be r-primitive if ord((alpha )=frac{q-1}{r}). In this paper, we discuss the existence of r-primitive pairs ((alpha , f(alpha ))) where (alpha in {mathbb {F}}_q), f(x) is a general rational function of degree sum m (degree sum is the sum of the degrees of numerator and denominator of f(x)) and the denominator of f(x) is square-free. Then we show that for any integer (m>0), there exists a positive constant (B_{r,m}) such that if (q>B_{r,m}), then such r-primitive pairs exist. In particular, we present a bound for (B_{r,m}) with (r=2) and (min {2,3,4,5,6}), and provide some conditions on the existence of 2-primitive pairs.

We prove that the Pommaret-Seiler resolution for quasi-stable ideals is cellular and give a cellular structure for it. This shows that this resolution is a generalization of the well known Eliahou–Kervaire resolution for stable ideals in a deeper sense. We also prove that the Pommaret-Seiler resolution can be reduced to the minimal one via Discrete Morse Theory and provide a constructive algorithm to perform this reduction.

In this paper, we consider smooth cubic surfaces with 15 lines. It is known that such surfaces can be generated by means of a double six with two pairs of Galois conjugate lines defined over the quadratic extension. The approach taken here is to consider the generation by means of a set of 9 lines defined over the field of coordinates. Eight lines arise from the double six, while the ninth is the diagonal line of the two pairs of Galois conjugate lines. This allows us to express all necessary equations and objects in terms of a set of four parameters over the coordinate field. As an application, we classify the smooth cubic surfaces with 15 lines over small finite fields by computer. All our results match with an enumerative formula recently found by Das.

Low-hit-zone frequency hopping sequence (LHZ-FHS) sets having optimal Hamming correlation are desirable in quasi-synchronous communication systems. In this paper, we first derive a new bound on maximum nontrivial Hamming correlation of LHZ-FHS sets from the famous Singleton bound in error correcting code literature, then obtain a general construction of LHZ-FHS sets from cyclic codes. Especially, two classes of LHZ-FHS sets meeting the new bound are constructed from punctured Reed-Solomon codes.

Let (Gamma (L, g)) be a Goppa code over ({mathbb {F}}_q), where (Lsubset mathbb {F}_{q^{m}}) is a support and (g(x)in mathbb {F}_{q^{m}}[x]) is a polynomial with s distinct roots in ({mathbb {F}}_{q^m}). In [Couvreur A, Otmani A, Tillich JP (2014) New identities relating wild Goppa codes. Finite Field Appl 29: 178–197.], Couvreur at al. gave the bound: (dim _{{mathbb {F}}_{q}}Gamma (L,g^e)-dim _{{mathbb {F}}_{q}}Gamma (L,g^{e+1})le s,) where (e=q^{m-1}+q^{m-2}+cdots +q). In this paper, we give the conditions such that (dim _{{mathbb {F}}_{q}}Gamma (L,g^e)=dim _{{mathbb {F}}_{q}}Gamma (L,g^{e+1})).

Kim, Chung, No and Chung introduced new families of M-ary sequences by using Sidel’nikov sequences and the shift-and-add method. Chung, No and Chung constructed more families of M-ary sequences based on Sidel’nikov sequences and the shift-and-inverse method. In this paper we further study the balancedness of these sequences and show that they have asymptotical uniform pattern distributions under some assumptions on the parameters of the sequences. Linear complexity profiles of the sequences are also studied.