Designs, Codes and Cryptography最新文献

The circular external difference family and its strong version are of great significance both in theory and in applications. In this paper, we apply the classical cyclotomic construction to the circular external differnece family and exhibit several concrete examples, in particular constructing an infinite family. Furthermore, we prove that all strong circular external differnece families are constructed by patching together several strong external difference families consisting of two subsets, thereby solving the open problem raised by Veitch and Stinson. We also present a new result on the non-existence of a certain type of strong external difference families.

We study the sequences generated by prefer-one rule with different initial vectors. Firstly, we give upper bounds of their periods and for initial vectors with Hamming weight one, we prove that the generated sequences are modified de Bruijn sequences. Moreover, for two of them, we give the truth tables of their feedback functions. We also investigate the feedback functions of prefer-one de Bruijn sequences. For order n prefer-one de Bruijn sequence, we give linear and quadratic terms in its feedback function and prove that the number of degree (n-2) terms has the same parity as n. The statistical result for small n shows that about half of all terms occur in the feedback functions.

We introduce a formal framework to study the multiple unicast problem for a coded network in which the network code is linear over a finite field and fixed. We show that the problem corresponds to an interference alignment problem over a finite field. In this context, we establish an outer bound for the achievable rate region and provide examples of networks where the bound is sharp. We finally give evidence of the crucial role played by the field characteristic in the problem.

Two skew Hadamard matrices are considered SH-equivalent if they are similar by a signed permutation matrix. This paper determines the number of SH-inequivalent skew Hadamard matrices of order 36 for some types. We also study ternary self-dual codes and association schemes constructed from the skew Hadamard matrices of order 36.

For projective Reed–Muller-type codes we give a global duality criterion in terms of the v-number and the Hilbert function of a vanishing ideal. As an application, we provide a global duality theorem for projective Reed–Muller-type codes over Gorenstein vanishing ideals, generalizing the known case where the vanishing ideal is a complete intersection. We classify self dual Reed–Muller-type codes over Gorenstein ideals using the regularity and a parity check matrix. For projective evaluation codes, we give a duality theorem inspired by that of affine evaluation codes. We show how to compute the regularity index of the r-th generalized Hamming weight function in terms of the standard indicator functions of the set of evaluation points.

Let (mathbb {F}_{q^n}) represent the finite field with (q^n) elements. In this paper, our focus is on determining the number of (mathbb {F}_{q^n})-rational points for two specific objects: an affine Artin–Schreier curve given by the equation (y^q-y = x(x^{q^i}-x)-lambda ), and an Artin–Schreier hypersurface given by the equation (y^q-y=sum _{j=1}^r a_jx_j(x_j^{q^{i_j}}-x_j)-lambda ). Additionally, we establish that the Weil bound is only achieved in these cases when the trace of the element (lambda in mathbb {F}_{q^n}) over the subfield (mathbb {F}_q) is equal to zero.

A Hadamard matrix is a scaled orthogonal matrix with (pm 1) entries. Such matrices exist in certain dimensions: the Hadamard conjecture is that such a matrix always exists when n is a multiple of 4. A conjecture attributed to Ryser is that no circulant Hadamard matrices exist when (n > 4). Recently, Dong and Rudelson proved the existence of approximate Hadamard matrices in all dimensions: there exist universal (0< c< C < infty ) so that for all (n ge 1), there is a matrix (A in left{ -1,1right} ^{n times n}) satisfying, for all (x in mathbb {R}^n),

We observe that, as a consequence of the existence of flat Littlewood polynomials, circulant approximate Hadamard matrices exist for all (n ge 1).

Online authenticated encryption has been considered of practical relevance in light-weight environments due to low latency and constant memory usage. In this paper, we propose a new tweakable block cipher-based online authenticated encryption scheme, dubbed ZLR, and its domain separation variant, dubbed DS-ZLR. ZLR and DS-ZLR follow the Encrypt-Mix-Encrypt paradigm. However, in contrast to existing schemes using the same paradigm such as ELmE and CoLM, ZLR and DS-ZLR enjoy n-bit security by using larger internal states with an efficient ZHash-like hashing algorithm. In this way, 2n-bit blocks are processed with only a single primitive call for hashing and two primitive calls for encryption and decryption, when they are based on an n-bit tweakable block cipher using n-bit (resp. 2n-bit) tweaks for ZLR (resp. DS-ZLR). Furthermore, they support pipelined computation as well as online nonce-misuse resistance. To the best of our knowledge, ZLR and DS-ZLR are the first pipelineable tweakable block cipher-based online authenticated encryption schemes of rate-2/3 that provide n-bit security with online nonce-misuse resistance.

Strongly multimedia identifiable parent property code (t-SMIPPC) was introduced for protecting multimedia contents from illegally redistributing under the averaging collusion attack. Such a code has efficient algorithms for tracing colluders. However, there are few results about the existence of such codes up to now. In this paper, we focus on t-SMIPPCs with length (t+1) where (t ge 2) is an integer. We first improve the lower bound on the size of such codes. For the case (t=2), i.e., 2-SMIPPCs with length 3, we further investigate combinatorial properties of the codes. Based on these properties, optimal q-ary 2-SMIPPCs with length 3 are constructed for (qequiv 0,1,2,5 pmod 6).

In this article we mainly study linear codes over ({mathbb {F}}_{2^n}) and their binary subfield codes. We construct linear codes over ({mathbb {F}}_{2^n}) whose defining sets are the certain subsets of ({mathbb {F}}_{2^n}^m) obtained from mathematical objects called simplicial complexes. We will consider simplicial complexes with one maximal element. The relation of the weights of codewords in two special codes obtained from simplicial complexes is illustrated by using LFSR sequences. And then we determine the parameters of these codes with the help of Boolean functions. As a result, we obtain five infinite families of distance optimal codes and give sufficient conditions for these codes to be minimal.