Advances in Mathematics of Communications最新文献

Let begin{document}$ G_2(q) $end{document} be a Chevalley group of type begin{document}$ G_2 $end{document} over a finite field begin{document}$ mathbb{F}_q $end{document}. Considering the begin{document}$ G_2(q) $end{document}-primitive action of rank begin{document}$ 3 $end{document} on the set of begin{document}$ frac{q^3(q^3-1)}{2} $end{document} hyperplanes of type begin{document}$ O_{6}^{-}(q) $end{document} in the begin{document}$ 7 $end{document}-dimensional orthogonal space begin{document}$ {{rm{PG}}}(7, q) $end{document}, we study the designs, codes, and some related geometric structures. We obtained the main parameters of the codes, the full automorphism groups of these structures, and geometric descriptions of the classes of minimum weight codewords.

In this paper we consider two pointsets in begin{document}$ mathrm{PG}(2,q^n) $end{document} arising from a linear set begin{document}$ L $end{document} of rank begin{document}$ n $end{document} contained in a line of begin{document}$ mathrm{PG}(2,q^n) $end{document}: the first one is a linear blocking set of Rédei type, the second one extends the construction of translation KM-arcs. We point out that their intersections pattern with lines is related to the weight distribution of the considered linear set begin{document}$ L $end{document}. We then consider the Hamming metric codes associated with both these constructions, for which we can completely describe their weight distributions. By choosing begin{document}$ L $end{document} to be an begin{document}$ {mathbb F}_{q} $end{document}-linear set with a short weight distribution, then the associated codes have few weights. We conclude the paper by providing a connection between the begin{document}$ Gammamathrm{L} $end{document}-class of begin{document}$ L $end{document} and the number of inequivalent codes we can construct starting from it.

MDS matrices play an important role in the design of block ciphers, and constructing MDS matrices with fewer xor gates is of significant interest for lightweight ciphers. For this topic, Duval and Leurent proposed an approach to construct MDS matrices by using three linear operations in ToSC 2018. Taking words as elements, they found begin{document}$ 16times16 $end{document} and begin{document}$ 32times 32 $end{document} MDS matrices over begin{document}$ mathbb{F}_2 $end{document} with only begin{document}$ 35 $end{document} xor gates and begin{document}$ 67 $end{document} xor gates respectively, which are also the best known implementations up to now. Based on the same observation as their work, we consider three linear operations as three kinds of elementary linear operations of matrices, and obtain more MDS matrices with begin{document}$ 35 $end{document} and begin{document}$ 67 $end{document} xor gates. In addition, some begin{document}$ 16times16 $end{document} or begin{document}$ 32times32 $end{document} involutory MDS matrices with only begin{document}$ 36 $end{document} or begin{document}$ 72 $end{document} xor gates over begin{document}$ mathbb{F}_2 $end{document} are also proposed, which are better than previous results. Moreover, our method can be extended to general linear groups, and we prove that the lower bound of the sequential xor count based on words for begin{document}$ 4 times 4 $end{document} MDS matrix over general linear groups is begin{document}$ 8n+2 $end{document}.