r -th order nonlinearity, correlation measure and least significant bit of the discrete logarithm Abstract Each finite binary sequence (sh) is associated with a Boolean function B. The correlation measure of order k and the r-th order nonlinearity are figures of merit for the unpredictability of (sh) and B, respectively. We estimate the r-th order nonlinearity of B in terms of the correlation measure of order 2r of (sh). We apply our result to Boolean functions associated with the Legendre sequence, that is, the binary sequence describing the least significant bit of the discrete logarithms in the finite field \(\mathbb {F}_{p}\) of pelements, where p > 2 is a prime.

A concatenated construction of linear complementary pair of codes Abstract A concatenated construction for linear complementary dual codes was given by Carlet et al. using the so-called isometry inner codes. Here, we obtain a concatenated construction to the more general family, linear complementary pair of codes. Moreover, we extend the dual code description of Chen et al. for concatenated codes to duals of generalized concatenated codes. This allows us to use generalized concatenated codes for the construction of linear complementary pair of codes.

R-2 composition tests: a family of statistical randomness tests for a collection of binary sequences Abstract In this article a family of statistical randomness tests for binary strings are introduced, based on Golomb’s pseudorandomness postulate R-2 on the number of runs. The basic idea is to construct recursive formulae with computationally tenable probability distribution functions. The technique is illustrated on testing strings of \(2^{7}\) , \(2^{8}\) , \(2^{10}\) and \(2^{12}\) bits. Furthermore, the expected value of the number of runs with a specific length is obtained. Finally the tests are applied to several collections of strings arising from different pseudorandom number generators.

New classes of quantum codes on closed orientable surfaces Abstract In this paper we construct two classes of binary quantum error-correcting codes on closed orientable surfaces. These codes are derived from self-dual orientable embeddings of complete bipartite graphs and complete multipartite graphs on the corresponding closed orientable surfaces. We also show a table comparing the rate of these quantum codes when fixing the minimum distance to 3 and 4.

A novel maximum distance separable code to generate universal identifiers Abstract Nowadays it is vital to have a robust mechanism that can identify people, objects, animals, and living beings, for example, for agricultural, health and national security purposes. Some drawbacks occur when very many objects need to be identified, and the tool is unable to support all of them. Even if the mechanism could tag them all, it is also important that the labels or codewords not resemble each other, to be able to detect and correct errors. To solve this problem, this article proposes an MDS (Maximum Distance Separable) code C with length 11 and dimension 7 over the finite field \({\mathbb {F}}_{2^{10}}\) . Furthermore, we construct a subcode of C with capacity for 327 different identifiers. Concretely we consider the set of all codewords with entries belonging to the subfield of \({\mathbb {F}}_{2^{10}}\) isomorphic to \({\mathbb {F}}_{2^{5}}\) . A decoding algorithm and an encryption method using elliptic curves cryptography for the codewords are also proposed.

A construction of Abelian non-cyclic orbit codes Abstract A constant dimension code consists of a set of k-dimensional subspaces of \(\mathbb {F}_{q}^{n}\) , where \(\mathbb {F}_{q}\) is a finite field of q elements. Orbit codes are constant dimension codes which are defined as orbits under the action of a subgroup of the general linear group on the set of all k-dimensional subspaces of \(\mathbb {F}_{q}^{n}\) . If the acting group is Abelian, we call the corresponding orbit code Abelian orbit code. In this paper we present a construction of an Abelian non-cyclic orbit code for which we compute its cardinality and its minimum subspace distance. Our code is a partial spread and consequently its minimum subspace distance is maximal.

New explicit injective compressing mappings on primitive sequences over ℤ p e $\mathbb {Z}_{p^e}$ Abstract Linear feedback shift registers over residue rings play a vital role in communication theory and cryptography. Let p be an odd prime and e ≥ 2 an integer. For any integer N ≥ 2, let \(\mathbb {Z}_{N}\) denote the residue ring modulo N. Let σ(x) be a primitive polynomial over \(\mathbb {Z}_{p^e}\) , and G′(σ(x),pe) the set of primitive linear recurring sequences generated by σ(x). Given a mapping \(\varphi :\mathbb {Z}_{p^e}\rightarrow \mathbb {Z}_{N}\) , its induced mapping \(\widehat {\varphi }\) transforms a sequence (…,ui− 1,ui,ui+ 1,… ) to (…,φ(ui− 1),φ(ui),φ(ui+ 1),… ). Then φ is called an injective compressing mapping (w.r.t. s-uniformity) if for any two distinct sequences \(\underline {u},\underline {v}\in G^{\prime }(\sigma (x),p^e)\) , at least one element of \(\mathbb {Z}_{N}\) ( \(s\in \mathbb {Z}_{N}\) ) is distributed in \(\widehat {\varphi }(\underline {u})\) differently from in \(\widehat {\varphi }(\underline {v})\) . It has been desirable to construct explicit injective compressing mappings (w.r.t. s-uniformity). Let the i-th coordinate ai of \(a\in \mathbb {Z}_{p^e}\) be given by \(a={\sum }_{i = 0}^{e-1}a_{i}p^{i}\) , \(a_{i}\in \mathbb {Z}_{p}\) . In this correspondence, it is proved that any permutation polynomial in the (e − 1)-th coordinate is an injective compressing mapping w.r.t. s-uniformity for all (but one) \(s\in \mathbb {Z}_{p}\) , and the efficiently implemented bitwise right-shift operator is an injective compressing mapping. Furthermore, two families of new injective compressing mappings are given in the form of coordinate polynomials.

Regular p -ary bent functions with five terms and Kloosterman sums Abstract Kloosterman sums are vital in the study of bent functions, including regular p-ary bent functions. In this paper, a congruence property for Kloosterman sums is presented first and is used to prove the nonexistence of a class of p-ary bent functions. Further, this paper considers p-ary functions of the form \(f(x)= \text {Tr}^{n}_{1}(a_{1}x^{r_{1}(q-1)})+\text {Tr}^{n}_{1}\left (c_{1}x^{r_{1}(q-1)+\frac {q^{2}-1}{2}}\right ) +\text {Tr}^{n}_{1}\left (a_{2}x^{r_{2}(q-1)}\right )+\text {Tr}^{n}_{1}\left (c_{2}x^{r_{2}(q-1)+\frac {q^{2}-1}{2}}\right ) +bx^{\frac {q^{2}-1}{2}}\) . We use Kloosterman sums in the characterization of this class of p-ary bent functions. Finally, an open problem of Jia et al. (IEEE Trans Inf. Theory 58(9): 6054–6063, 2012) is solved and we prove the nonexistence for a class of regular p-ary bent functions.

On Boolean functions with several flat spectra Abstract In this paper, two constructions of Boolean functions which have at least two flat spectra with respect to {H, N}n are proposed. Some known results about bent-negabent functions can be seen as special cases of our results. Furthermore, some lower bounds on the numbers of flat spectra of Boolean functions with respect to {H, N}n or {I, N}n are given. In particular, we show that any Maiorana-McFarland bent function of n (n even) variables has at least \(\frac {n}{2}+ 2^{\frac {n}{2}} \) flat spectra with respect to {H, N}n. Finally, following the work by Riera, Petrides, and Parker, we develop recursive formulae for the numbers of flat spectra of some structural quadratics, including star function, star-line function, and star-line-star function.

New bounds for linear codes of covering radii 2 and 3 Abstract The length function ℓq(r, R) is the smallest length of a q-ary linear code of covering radius R and codimension r. In this work we obtain new upper bounds on ℓq(2t + 1,2), ℓq(3t + 1,3), ℓq(3t + 2,3), t ≥ 1. In particular, we prove that $$\ell_{q}(3,2)\le\sqrt{q(3\ln q+\ln\ln q)}+\sqrt{\frac{q}{3\ln q}}+ 3~\text{ for all } q, $$ \(\ell _{q}(3,2)\le 1.05\sqrt {3q\ln q}\) for q ≤ 321007, \(\ell _{q}(4,3)<2.8\sqrt [3]{q\ln q}\) for q ≤ 6229, and \(\ell _{q}(5,3)<3\sqrt [3]{q^{2}\ln q}\) for q ≤ 761. The new bounds on ℓq(2t + 1,2), ℓq(3t + 1,3), ℓq(3t + 2,3), t > 1, are then obtained by lift-constructions. For q a non-square the new bound on ℓq(2t + 1,2) improves the previously known ones. For many values of q≠(q′)3 and r ≠ 3t we provide infinite families of [n, n − r]q3 codes showing that \(\ell _{q}(r,3)\thickapprox c\sqrt [3]{\ln q}\cdot q^{(r-3)/3}\) , where c is a universal constant. As far as it is known to the authors, such families have not been previously described in the literature.