Applicable Algebra in Engineering Communication and Computing最新文献

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.

In this paper we study the problem of constructing non-trivial subtowers and supertowers of recursive towers of function fields over finite fields.

We introduce the concept of subalgebra spectrum, Sp(A), for a subalgebra A of finite codimension in (mathbb {K}[x]). The spectrum is a finite subset of the underlying field. We also introduce a tool, the characteristic polynomial of A, which has the spectrum as its set of zeroes. The characteristic polynomial can be computed from the generators of A, thus allowing us to find the spectrum of an algebra given by generators. We proceed by using the spectrum to get descriptions of subalgebras of finite codimension. More precisely we show that A can be described by a set of conditions that each is either of the type (f(alpha )=f(beta )) for (alpha ,beta) in Sp(A) or of the type stating that some linear combination of derivatives of different orders evaluated in elements of Sp(A) equals zero. We use these types of conditions to, by an inductive process, find explicit descriptions of subalgebras of codimension up to three. These descriptions also include SAGBI bases for each family of subalgebras.

This paper presents two modifications for Loidreau’s cryptosystem, a rank metric-based cryptosystem constructed by using Gabidulin codes in the McEliece setting. Recently a polynomial-time key recovery attack was proposed to break this cryptosystem in some cases. To prevent this attack, we propose the use of subcodes to disguise the secret codes in Modification I. In Modification II, we choose a random matrix of low column rank to mix with the secret matrix. Our analysis shows that these two modifications can both resist the existing structural attacks. Furthermore, these modifications have a much more compact representation of public keys compared to Classic McEliece, which has been selected into the fourth round of the NIST-PQC project.