Advances in Mathematics of Communications最新文献

Recently, the construction of new MDS Euclidean self-dual codes has been widely investigated. In this paper, for square begin{document}$ q $end{document}, we utilize generalized Reed-Solomon (GRS) codes and their extended codes to provide four generic families of begin{document}$ q $end{document}-ary MDS Euclidean self-dual codes of lengths in the form begin{document}$ sfrac{q-1}{a}+tfrac{q-1}{b} $end{document}, where begin{document}$ s $end{document} and begin{document}$ t $end{document} range in some interval and begin{document}$ a, b ,|, (q -1) $end{document}. In particular, for large square begin{document}$ q $end{document}, our constructions take up a proportion of generally more than 34% in all the possible lengths of begin{document}$ q $end{document}-ary MDS Euclidean self-dual codes, which is larger than the previous results. Moreover, two new families of MDS Euclidean self-orthogonal codes and two new families of MDS Euclidean almost self-dual codes are given similarly.

Let begin{document}$ mgeq3 $end{document} be a positive integer and begin{document}$ n = 2m $end{document}. Let begin{document}$ f(x) = x^{2^m+3} $end{document} be a power permutation over begin{document}$ {mathrm {GF}}(2^n) $end{document}, which is a monomial with a Niho exponent. In this paper, the differential spectrum of begin{document}$ f $end{document} is investigated. It is shown that the differential spectrum of begin{document}$ f $end{document} is begin{document}$ mathbb S = {omega_0 = 2^{2m-1}+2^{2m-3}-1,omega_2 = 2^{2m-2}+2^{m-1}, omega_4 = 2^{2m-3}-2^{m-1},omega_{2^m} = 1} $end{document} when begin{document}$ m $end{document} is even, and begin{document}$ mathbb S = {omega_0 = frac{7cdot2^{2m-2}+2^m}3, omega_2 = 3cdot2^{2m-3}-2^{m-2}-1, omega_6 = frac{2^{2m-3}-2^{m-2}}3, omega_{2^m+2} = 1} $end{document} when begin{document}$ m $end{document} is odd.

Z-complementary pairs (ZCPs) have been widely used in different communication systems. In this paper, we first investigate the odd-periodic correlation property of ZCPs, and propose a new class of ZCPs, called ZOC-ZCPs with zero correlation zone (ZCZ) width begin{document}$ Z $end{document} and zero odd-period correlation zone (ZOCZ) width begin{document}$ Z_{odd} = Z $end{document} by horizontal concatenation of a certain combination of some known ZCPs. Particularly, based on any known Golay pair, we can generate a class of GCPs of more flexible length whose ZOCZ width is larger than a quarter of the sequence length.

We study the number begin{document}$ R_n(t,N) $end{document} of tuplets begin{document}$ (x_1,ldots, x_n) $end{document} of congruence classes modulo begin{document}$ N $end{document} such that

begin{document}$ begin{equation*} x_1cdots x_n equiv t pmod{N}. end{equation*} $end{document}

As a result, we derive a recurrence for begin{document}$ R_n(t,N) $end{document} and prove some multiplicative properties of begin{document}$ R_n(t,N) $end{document}. Furthermore, we apply the result to study the probability distribution of Diffie-Hellman keys used in multiparty communication. We show that this probability distribution is not uniform.

Permutation codes under the Hamming metric are interesting topics due to their applications in power line communications and block ciphers. In this paper, we study perfect permutation codes in begin{document}$ S_n $end{document}, the set of all permutations on begin{document}$ n $end{document} elements, under the Hamming metric. We prove the nonexistence of perfect begin{document}$ t $end{document}-error-correcting codes in begin{document}$ S_n $end{document} under the Hamming metric, for more values of begin{document}$ n $end{document} and begin{document}$ t $end{document}. Specifically, we propose some sufficient conditions of the nonexistence of perfect permutation codes. Further, we prove that there does not exist a perfect begin{document}$ t $end{document}-error-correcting code in begin{document}$ S_n $end{document} under the Hamming metric for some begin{document}$ n $end{document} and begin{document}$ t = 1,2,3,4 $end{document}, or begin{document}$ 2t+1leq nleq max{4t^2e^{-2+1/t}-2,2t+1} $end{document} for begin{document}$ tgeq 2 $end{document}, or begin{document}$ min{frac{e}{2}sqrt{n+2},lfloorfrac{n-1}{2}rfloor}leq tleq lfloorfrac{n-1}{2}rfloor $end{document} for begin{document}$ ngeq 7 $end{document}, where begin{document}$ e $end{document} is the Napier's constant.