On degree- d zero-sum sets of full rank Abstract A set \(S \subseteq {{\mathbb {F}}_{2}^{n}}\) is called degree-d zero-sum if the sum \({\sum }_{s \in S} f(s)\) vanishes for all n-bit Boolean functions of algebraic degree at most d. Those sets correspond to the supports of the n-bit Boolean functions of degree at most n − d − 1. We prove some results on the existence of degree-d zero-sum sets of full rank, i.e., those that contain n linearly independent elements, and show relations to degree-1 annihilator spaces of Boolean functions and semi-orthogonal matrices. We are particularly interested in the smallest of such sets and prove bounds on the minimum number of elements in a degree-d zero-sum set of rank n. The motivation for studying those objects comes from the fact that degree-d zero-sum sets of full rank can be used to build linear mappings that preserve special kinds of nonlinear invariants, similar to those obtained from orthogonal matrices and exploited by Todo, Leander and Sasaki for breaking the block ciphers Midori, Scream and iScream.

One construction of perfect ternary sequences Abstract We present a new construction of a family of perfect ternary sequences (PTSs) that is a generalization of one of the known families of PTSs. These PTSs of length N1N2 are derived from shift sequences of odd length N1 corresponding to m-sequences over GF(p) and PTSs of odd length N2. Ipatov PTSs are a special case where N2 = 1. For N2 ≥ 3, we find conditions under which the obtained PTSs are new. We also consider implementation issues of these sequences.

The 2-adic complexity of a class of binary sequences with optimal autocorrelation magnitude Abstract Recently, a class of binary sequences with optimal autocorrelation magnitude has been presented by Su et al. based on Ding-Helleseth-Lam sequences and interleaving technique (Designs, Codes and Cryptography 86, 1329–1338, 2018). The linear complexity of this class of sequences has been proved to be large enough to resist the B-M Algorithm by Fan (Designs, Codes and Cryptography 86, 2441–2450, 2018). In this paper, we study the 2-adic complexities of these sequences with period 4p and show they are no less than 2p, i.e., its 2-adic complexity is large enough to resist the Rational Approximation Algorithm.

On the number of the rational zeros of linearized polynomials and the second-order nonlinearity of cubic Boolean functions Abstract Determine the number of the rational zeros of any given linearized polynomial is one of the vital problems in finite field theory, with applications in modern symmetric cryptosystems. But, the known general theory for this task is much far from giving the exact number when applied to a specific linearized polynomial. The first contribution of this paper is a better general method to get a more precise upper bound on the number of rational zeros of any given linearized polynomial over arbitrary finite field. We anticipate this method would be applied as a useful tool in many research branches of finite field and cryptography. Really we apply this result to get tighter estimations of the lower bounds on the second-order nonlinearities of general cubic Boolean functions, which has been an active research problem during the past decade. Furthermore, this paper shows that by studying the distribution of radicals of derivatives of a given Boolean function one can get a better lower bound of the second-order nonlinearity, through an example of the monomial Boolean functions \(g_{\mu }=Tr(\mu x^{2^{2r}+2^{r}+1})\) defined over the finite field \({\mathbb F}_{2^{n}}\).

Two constructions for 16-QAM complementary sequence sets with non-power-of-two length Abstract Complementary sequences with quadrature amplitude modulation (QAM) symbols have important applications in OFDM communication systems. The objective of this paper is to present two constructions of 16-QAM complementary sequence sets of size 4. The first construction generates four complementary sequences of length L = 2m− 1 + 2v, where m and v are two positive integers with 1 ≤ v ≤ m − 1. The second one leads to four complementary sequences of length L = 2m− 1 + 1. It turns out that the peak-to-mean envelope power ratios (PMEPRs) of constructed complementary sequence sets are upper bounded by 4.

A note on the minimal binary linear code Abstract Due to the wide applications in communications, data storage and cryptography, linear codes have received much attention in the past decades. As a subclass of linear codes, minimal linear codes can be used to construct secret sharing with nice access structure. The objective of this paper is to construct new classes of minimal binary linear codes with \(w_{\min \limits }/w_{\max \limits }\leq 1/2\) from preferred binary linear codes, where \(w_{\min \limits }\) and \(w_{\max \limits }\) denote the minimum and maximum nonzero Hamming weights in \(\mathcal {C}\) respectively. Firstly, we introduce a concept called preferred binary linear codes and a class of minimal binary linear codes with \(w_{\min \limits }/w_{\max \limits }\leq 1/2\) can be deduced from preferred binary linear codes. As an application of preferred binary linear codes, we get a new class of six-weight minimal binary linear codes with \(w_{\min \limits }/w_{\max \limits }< 1/2\) from a known class of five-weight preferred binary linear codes. Secondly, by employing vectorial Boolean functions, we construct two new classes of preferred binary linear codes and, consequently, these two new classes of preferred binary linear codes can generate two new classes of minimal binary linear codes with \(w_{\min \limits }/w_{\max \limits }\leq 1/2\) and large minimum distance.

New quaternary sequences with optimal odd-periodic autocorrelation magnitude Abstract A quaternary sequence is said to be optimal if its odd-periodic autocorrelation magnitude equal to 2 for even length, and 1 for odd length. In this paper, we propose three constructions of optimal quaternary sequences: the first construction applies the inverse Gray mapping to four component binary sequences, which could be chosen from GMW sequence pair, twin-prime sequence pair, Legendre sequence pair, and ideal sequences; the second one generates optimal sequences from quaternary sequences with optimal even-periodic autocorrelation magnitude; the third one gives new optimal quaternary sequences by applying the sign alternation transform and Gray mapping to GMW sequence pair and twin-prime sequence pair. In particular, some proposed sequences have new parameters.

A class of constacyclic BCH codes Abstract Constacyclic codes are a subclass of linear codes and have been well studied. Constacyclic BCH codes are a family of constacyclic codes and contain BCH codes as a subclass. Compared with the in-depth study of BCH codes, there are relatively little study on constacyclic BCH codes. The objective of this paper is to determine the dimension and minimum distance of a class of q-ary constacyclic BCH codes of length \(\frac {q^{m}-1}{q-1}\) with designed distances \(\delta _{i}=q^{m-1}-\frac {q^{\lfloor \frac {m-3}2 \rfloor +i }-1}{q-1}\) for \(1\leq i\leq \min \limits \{\lceil \frac {m+1}2 \rceil -\lfloor \frac {m}{q+1} \rfloor , \lceil \frac {m-1}2 \rceil \}\) . As will be seen, some of these codes are optimal.

Editorial: Special issue on Boolean functions and their applications 2018

Image sets with regularity of differences Abstract In the last few decades we’ve seen several results connecting the image sets of some special functions to differences sets and partial difference sets. Examples here include planar functions (skew-Hadamard difference sets), and (more classically) monomials (cyclotomic DS). It can be observed that there is a commonality (in the main) among the behaviour of these functions, in that there tends to be a certain regularity in the number of times each image occurs. In this paper, we instigate a new approach to constructing sets with regularity of differences based on the above observation. Specifically, we show how functions over finite fields exhibiting a regularity of images can yield image sets that exhibit some sort of difference regularity.