American Institute of Mathematical Sciences

Let \begin{document}$\mathbb{F}_{p^m}$\end{document} be a finite field of cardinality \begin{document}$p^m$\end{document} and \begin{document}$R = \mathbb{F}_{p^m}[u]/\langle u^2\rangle = \mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$\end{document} \begin{document}$(u^2 = 0)$\end{document}, where \begin{document}$p$\end{document} is a prime and \begin{document}$m$\end{document} is a positive integer. For any \begin{document}$λ∈ \mathbb{F}_{p^m}^{×}$\end{document}, an explicit representation for all distinct \begin{document}$λ$\end{document}-constacyclic codes over \begin{document}$R$\end{document} of length \begin{document}$np^s$\end{document} is given by a canonical form decomposition for each code, where \begin{document}$s$\end{document} and \begin{document}$n$\end{document} are arbitrary positive integers satisfying \begin{document}${\rm gcd}(p,n) = 1$\end{document}. For any such code, using its canonical form decomposition the representation for the dual code of the code is provided. Moreover, representations for all distinct cyclic codes, negacyclic codes and their dual codes of length \begin{document}$np^s$\end{document} over \begin{document}$R$\end{document} are obtained, and self-duality for these codes are determined. Finally, all distinct self-dual negacyclic codes over \begin{document}$\mathbb{F}_5+u\mathbb{F}_5$\end{document} of length \begin{document}$2· 3^t· 5^s$\end{document} are listed for any positive integer \begin{document}$t$\end{document}.

Recently, several striking advances have taken place regarding the discrete logarithm problem (DLP) in finite fields of small characteristic, despite progress having remained essentially static for nearly thirty years, with the best known algorithms being of subexponential complexity. In this expository article we describe the key insights and constructions which culminated in two independent quasi-polynomial algorithms. To put these developments into both a historical and a mathematical context, as well as to provide a comparison with the cases of so-called large and medium characteristic fields, we give an overview of the state-of-the-art algorithms for computing discrete logarithms in all finite fields. Our presentation aims to guide the reader through the algorithms and their complexity analyses ab initio.

We present an application of Hilbert quasi-polynomials to order domains, allowing the effective check of the second order-domain condition in a direct way. We also provide an improved algorithm for the computation of the related Hilbert quasi-polynomials. This allows to identify order domain codes more easily.

As an optimal combinatorial object, bent functions have been an interesting research object due to their important applications in cryptography, coding theory, and sequence design. The characterization and construction of bent functions are challenging problems in general. The objective of this paper is to present a construction of p-ary weakly regular bent functions from known weakly regular bent functions. This generalizes some earlier constructions of Boolean bent functions and p-ary bent functions, and produces several infinite families of p-ary weakly regular bent functions from known ones. Some infinite families of p-ary rotation symmetric bent functions are obtained as well.

We generalize the construction of locally recoverable codes on algebraic curves given by Barg, Tamo and Vlăduţ [4] to those with arbitrarily many recovery sets by exploiting the structure of fiber products of curves. Employing maximal curves, we create several new families of locally recoverable codes with multiple recovery sets, including codes with two recovery sets from the generalized Giulietti and Korchmáros (GK) curves and the Suzuki curves, and new locally recoverable codes with many recovery sets based on the Hermitian curve, using a fiber product construction of van der Geer and van der Vlugt. In addition, we consider the relationship between local error recovery and global error correction as well as the availability required to locally recover any pattern of a fixed number of erasures.

We construct new families of completely regular codes by concatenation methods. By combining parity check matrices of cyclic Hamming codes, we obtain families of completely regular codes. In all cases, we compute the intersection array of these codes. As a result, we find some non-equivalent completely regular codes, over the same finite field, with the same parameters and intersection array. We also study when the extension of these codes gives completely regular codes. Some of these new codes are completely transitive.

In statistical planning of experiments, super-simple designs are the ones providing samples with maximum intersection as small as possible. Super-simple pairwise balanced designs are useful in constructing other types of super-simple designs which can be applied to codes and designs. In this paper, the super-simple pairwise balanced designs with block sizes 3 and 4 are investigated and it is proved that the necessary conditions for the existence of a super-simple \begin{document}$(v, \{3,4\}, λ)$\end{document}-PBD for \begin{document}$λ = 7,9$\end{document} and \begin{document}$λ = 2k$\end{document}, \begin{document}$k≥1$\end{document}, are sufficient with seven possible exceptions. In the end, several optical orthogonal codes and superimposed codes are given.

We investigate a family of codes called quasi-quadratic residue (QQR) codes. We are interested in these codes mainly for two reasons: Firstly, they have close relations with hyperelliptic curves and Goppa's Conjecture, and serve as a strong tool in studying those objects. Secondly, they are very good codes. Computational results show they have large minimum distances when \begin{document}$p\equiv 3 \pmod 8$\end{document}.

Our studies focus on the weight distributions of these codes. We will prove a new discovery about their weight polynomials, i.e. they are divisible by \begin{document}$(x^2 + y^2)^{d-1}$\end{document}, where \begin{document}$d$\end{document} is the corresponding minimum distance. We also show that the weight distributions of these codes are asymptotically normal. Based on the relation between QQR codes and hyperelliptic curves, we will also prove a result on the point distribution on hyperelliptic curves.

Frequency hopping sequences (FHSs) with favorable partial Hamming correlation properties have important applications in many synchronization and multiple-access systems. In this paper, we investigate constructions of FHS sets with optimal partial Hamming correlation. We present several direct constructions for balanced nested cyclic difference packings (BNCDPs) and balanced nested cyclic relative difference packings (BNCRDPs) by using trace functions and discrete logarithm. We also show three recursive constructions for FHS sets with partial Hamming correlation, which are based on cyclic difference matrices and discrete logarithm. Combing these BNCDPs, BNCRDPs and three recursive constructions, we obtain infinitely many new strictly optimal FHS sets with respect to the Peng-Fan bounds.

We generalize the code constructed recently by Wang et al, and obtain many classes of codes with a few weights. The weight distribution of these codes is completely determined, and the minimum distance of the duals of these codes is determined. We also show that some subclasses of the duals of these codes are optimal. Furthermore, some parameters of the generalized Hamming weight of these codes are calculated in certain cases.