Discrete Mathematics最新文献

The Szegedy quantum walk is a discrete time quantum walk model which defines a quantum analogue of any Markov chain. The long-term behavior of the quantum walk can be encoded in a matrix called the average mixing matrix, whose columns give the limiting probability distribution of the walk given an initial state. We define a version of the average mixing matrix of the Szegedy quantum walk which allows us to more readily compare the limiting behavior to that of the chain it quantizes. We prove a formula for our mixing matrix in terms of the spectral decomposition of the Markov chain and show a relationship with the mixing matrix of a continuous quantum walk on the chain. In particular, we prove that average uniform mixing in the continuous walk implies average uniform mixing in the Szegedy walk. We conclude by giving examples of Markov chains of arbitrarily large size which admit average uniform mixing in both the continuous and Szegedy quantum walk.

Based on the definition of Hamiltonian cycles by Katona and Kierstead, we present a recursive construction of tight Hamiltonian decompositions of complete 3-uniform hypergraphs $K_n^{\left(3\right)}$, and complete multipartite 3-uniform hypergraph $K_{t\left(n\right)}^{\left(3\right)}$, where t is the number of partite sets and n is the size of each partite set. For $t\equiv 4,8\left(\mathrm{mod}12\right)$, we utilize a tight Hamiltonian decomposition of $K_t^{\left(3\right)}$ to construct those of $K_{2t}^{\left(3\right)}$ and $K_{t\left(n\right)}^{\left(3\right)}$ for all positive integers n. By applying our method in conjunction with the current results in literature, we obtain tight Hamiltonian decompositions for infinitely many hypergraphs, namely complete hypergraphs $K_t^{\left(3\right)}$ and complete multipartite hypergraphs $K_{t\left(n\right)}^{\left(3\right)}$ for any positive integer n, and $t=2^m,5\cdot 2^m,7\cdot 2^m$, and $11\cdot 2^m$ when $m\ge 2$.

Linear complementary pair (abbreviated to LCP) of codes were defined by Ngo et al. in 2015, and were proved that these pairs of codes can help to improve the security of the information processed by sensitive devices, especially against so-called side-channel attacks (SCA) and fault injection attacks (FIA). In this paper, we first generalize the LCP of codes over finite fields to the additive complementary pair (ACP) of codes in the ambient space with mixed binary and quaternary alphabets. Then we provide two characterizations for the $Z_2{Z}_4$-additive codes pair $(C,D)$ to be $Z_2{Z}_4$-ACP of codes. Meanwhile, we obtain a sufficient condition for the $Z_2{Z}_4$-additive codes pair $(C,D)$ to be $Z_2{Z}_4$-ACP of codes. Under suitable conditions, we derive a necessary and sufficient condition for the Gray map Φ image of $Z_2{Z}_4$-ACP of codes $(C,D)$ to be LCP of codes over $Z_2$. Finally, we exhibit an interesting application of a special class of the $Z_2{Z}_4$-ACP of codes in coding for the two-user binary adder channel.

Linear complementary dual (LCD) codes and linear complementary pairs (LCP) of codes have been proposed for new applications as countermeasures against side-channel attacks (SCA) and fault injection attacks (FIA) in the context of direct sum masking (DSM). The countermeasure against FIA may lead to a vulnerability for SCA when the whole algorithm needs to be masked (in environments like smart cards). This led to a variant of the LCD and LCP problems, where several results were obtained intensively for LCD codes, but only partial results were derived for LCP codes. Given the gap between the thin results and their particular importance, this paper aims to reduce this by further studying the LCP of codes in special code families and, precisely, the characterization and construction mechanism of LCP codes of algebraic geometry codes over finite fields. Notably, we propose constructing explicit LCP of codes from elliptic curves. Besides, we also study the security parameters of the derived LCP of codes $(C,D)$ (notably for cyclic codes), which are given by the minimum distances $d\left(C\right)$ and $d\left(D^{\perp }\right)$. Further, we show that for LCP algebraic geometry codes $(C,D)$, the dual code $C^{\perp }$ is equivalent to $D$ under some specific conditions we exhibit. Finally, we investigate whether MDS LCP of algebraic geometry codes exist (MDS codes are among the most important in coding theory due to their theoretical significance and practical interests). Construction schemes for obtaining LCD codes from any algebraic curve were given in 2018 by Mesnager, Tang and Qi in [11]. To our knowledge, it is the first time LCP of algebraic geometry codes has been studied.