Let f(x) = xd be a power mapping over \(\mathbb {F}_{n}\) and \(\mathcal {U}_{d}\) the maximum number of solutions \(x\in \mathbb {F}_{n}\) of \({\Delta }_{f,c}(x):=f(x+c)-f(x)=a\text {, where }c,a\in \mathbb {F}_{n}\text { and } c\neq 0\). f is said to be differentially k-uniform if \(\mathcal {U}_{d} =k\). The investigation of power functions with low differential uniformity over finite fields \(\mathbb

Inspired by a recent work of Tang et al. on constructing bent functions [14, IEEE TIT, 63(1): 6149-6157, 2017], we introduce a property (Pτ) of any Boolean function that its second order derivatives vanish at any direction (ui,uj) for some τ-subset {u1,…,uτ} of \(\mathbb {F}_{2^{n}}\), and then establish a link between this property and the construction of Tang et al. (IEEE Trans. Inf. Theory 63(10)

Linear codes have a wide range of applications in the data storage systems, communication systems, consumer electronics products since their algebraic structure can be analyzed and they are easy to implement in hardware. How to construct linear codes with excellent properties to meet the demands of practical systems becomes a research topic, and it is an efficient way to construct linear codes from

We consider n-bit permutations with differential uniformity of 4 and null nonlinearity. We first show that the inverses of Gold functions have the interesting property that one component can be replaced by a linear function such that it still remains a permutation. This directly yields a construction of 4-uniform permutations with trivial nonlinearity in odd dimension. We further show their existence

Recently Budaghyan et al. (Cryptogr. Commun. 12, 85–100, 2020) introduced a procedure for investigating if CCZ-equivalence can be more general than EA-equivalence together with inverse transformation (when applicable). In this paper, we show that it is possible to use this procedure for classifying, up to EA-equivalence, all known APN functions in dimension 6. We also give some discussion for dimension

For a Distributed Storage System (DSS), the Fractional Repetition (FR) code is a class in which replicas of encoded data packets are stored on distributed chunk servers, where the encoding is done using the Maximum Distance Separable (MDS) code. The FR codes allow for the exact uncoded repair with minimum repair bandwidth. In this paper, FR codes (called Flower codes) are constructed using finite binary

Let \(\mathbb {F}_{2^{m}}\) be the finite field with 2m elements, where m is a positive integer. Recently, Heng and Ding in (Finite Fields Appl. 56:308–331, 2019) studied the subfield codes of two families of hyperovel codes and determined the weight distribution of the linear code $$ \mathcal{C}_{a,b}=\left\{((\text{Tr}_{1}^{m}(a f(x)+bx)+c)_{x \in \mathbb{F}_{2^{m}}}, \text{Tr}_{1}^{m}(a), \text{Tr}_{1}^{m}(b))

The set of linear structures of most known balanced Boolean functions is non-trivial. In this paper, some balanced Boolean functions whose set of linear structures is trivial are constructed. We show that any APN function in even dimension must have a component whose set of linear structures is trivial. We determine a general form for the number of bent components in quadratic APN functions in even

We correct some mistakes in the paper “A mass formula for negacyclic codes of length 2k and some good negacyclic codes over \(\mathbb {Z}_{4}+u\mathbb {Z}_{4}\)” (Bandi et al. Cryptogr. Commun. 9, 241–272, 2017).

A bent4 function is a Boolean function with a flat spectrum with respect to a certain unitary transform \(\mathcal {T}\). It was shown previously that a Boolean function f in an even number of variables is bent4 if and only if f + σ is bent, where σ is a certain quadratic function depending on \(\mathcal {T}\). Hence bent4 functions are also called shifted bent functions. Similarly, a Boolean function

The concept of a locally repairable code (LRC) was introduced to protect the data from disk failures in large-scale storage systems. In this paper, we consider the LRCs with multiple disjoint repair sets and each repair set contains exactly one check symbol. By using several structures from combinatorial design theory, such as balanced incomplete block design, cyclic packing, group divisible design

The original version of this article unfortunately contained a mistake in the main title.

Let \(R=\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle \). Then R is a local non-principal ideal ring of 16 elements. First, we give the structure of every cyclic code of odd length n over R and obtain a complete classification for these codes. Then we determine the cardinality, the type and its dual code for each of these cyclic codes. Moreover, we determine all self-dual cyclic codes of odd length n

In this paper, we investigate a family of q2-ary narrow-sense and non-narrow-sense negacyclic BCH codes with length \(n=\frac {q^{2m}-1}{2}\), where q is an odd prime power and m ≥ 3 is odd. We propose Hermitian dual-containing conditions for narrow-sense and non-narrow-sense negacyclic BCH codes, and precisely compute the dimensions of these negacyclic BCH codes whose maximal designed distance can

In this paper, we study the differential uniformity of the composition of two functions with the help of Boolean matrix theory. Based on the result of our research, we can construct new differentially 4-uniform permutations from known ones. In addition, we find some clues about the existence of APN permutations of \(\mathbb {F}_{2^{n}}\) for even n ≥ 8.

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

Let NFSR(f ) denote the nonlinear feedback shift register (NFSR) with characteristic function f = x0 ⊕ g(x1,x2,…,xn− 1) ⊕ xn. In this paper, the cycle structure of NFSR(fd) is discussed, where \(f^{d}=x_{0}\oplus g(x_{d},x_{2d},{\ldots } ,x_{(n-1)d})\oplus x_{nd}\) is also a characteristic function determined by f and a given integer d. If the cycle structure of NFSR(f ) is known, then it is shown

We construct a class of \(\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}\)-additive cyclic codes generated by pairs of polynomials, where p is a prime number. The generator matrix of this class of codes is obtained. By establishing the relationship between the random \(\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}\)-additive cyclic code and random quasi-cyclic code of index 2 over \(\mathbb {Z}_{p}\), the asymptotic properties

This paper presents a class of integer codes capable of correcting l-bit burst asymmetric errors within a b-bit byte (1 ≤ l < b) and double asymmetric errors within a codeword. The presented codes are constructed with the help of a computer and have the potential to be used in unamplified optical networks. In addition, the paper derives the upper bound on code length and shows that the proposed codes

Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. We study the largest minimum weights d(n,k) among all binary linear complementary dual [n,k] codes. We determine d(n,4) for n ≡ 2,3,4,5,6,9,10,13 (mod 15), and d(n,5) for n ≡ 3,4,5,7,11,19,20, 22,26 (mod 31). Combined with known results, d(n,k) are also determined for

Involutions over finite fields are permutations whose compositional inverses are themselves. Involutions especially over \( \mathbb {F}_{q} \) with q is even have been used in many applications, including cryptography and coding theory. The explicit study of involutions (including their fixed points) has started with the paper (Charpin et al. IEEE Trans. Inf. Theory, 62(4), 2266–2276 2016) for binary

In this paper we study almost p-ary sequences and their autocorrelation coefficients. We first study the number ℓ of distinct out-of-phase autocorrelation coefficients for an almost p-ary sequence of period n + s with s consecutive zero-symbols. We prove an upper bound and a lower bound on ℓ. It is shown that ℓ can not be less than \(\min \limits \{s,p,n\}\). In particular, it is shown that a nearly

This paper presents an explicit representation for the solutions of the equation \({\sum }_{i=0}^{\frac kl-1}x^{2^{li}} = a \in \mathbb {F}_{2^{n}}\) for any given positive integers k, l with l|k and n, in the closed field \({\overline {\mathbb {F}_{2}}}\) and in the finite field \(\mathbb {F}_{2^{n}}\). As a by-product of our study, we are able to completely characterize the a’s for which this equation

We consider a special type of sequence of arithmetic progressions, in which consecutive progressions are related by the property: ithterms ofjth, (j + 1)thprogressions of the sequence are multiplicative inverses of each other modulo(i + 1)thterm ofjthprogression. Such a sequence is uniquely defined for any pair of co-prime numbers. A computational problem, defined in the context of such a sequence

Sequence families with zero correlation zone (ZCZ) have been extensively studied in recent years due to their important applications in quasi-synchronous code-division multiple-access (QS-CDMA) systems. To accommodate multiuser environments, multiple ZCZ sequence sets with low inter-set cross-correlation are expected. In this paper, we propose a construction of polyphase ZCZ sequences based on generalised

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

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

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 \(w_{\min \limits }/w_{\max \limits }\leq 1/2\)

We present techniques to obtain small circuits which also have low depth. The techniques apply to typical cryptographic functions, as these are often specified over the field GF (2), and they produce circuits containing only AND, XOR and XNOR gates. The emphasis is on the linear components (those portions containing no AND gates). A new heuristic, DCLO (for depth-constrained linear optimization), is

A necessary condition for the security of cryptographic functions is to be "sufficiently distant" from linear, and cryptographers have proposed several measures for this distance. In this paper, we show that six common measures, nonlinearity, algebraic degree, annihilator immunity, algebraic thickness, normality, and multiplicative complexity, are incomparable in the sense that for each pair of measures

Depending on the parity of n and the regularity of a bent function f from F p n to F p , f can be affine on a subspace of dimension at most n/2, (n - 1)/2 or n/2 - 1. We point out that many p-ary bent functions take on this bound, and it seems not easy to find examples for which one can show a different behaviour. This resembles the situation for Boolean bent functions of which many are (weakly) n/2-normal

This paper generalizes three constructions of families of sequences with bounded off peak correlation with application to Code Division Multiple Access (CDMA), frequency hopping, and Ultra Wide Band (UWB). These new families present flexible family sizes and sequence lengths, making them well suited to wireless communications and Multiple Input Multiple Output (MIMO) radar. In particular, we show that

Based on the parity of the number of occurrences of a pattern 10 as a scattered subsequence in the binary representation of integers, a Rudin-Shapiro-like sequence is defined by Lafrance, Rampersad and Yee. The N th maximum order complexity and the expansion complexity of this Rudin-Shapiro-like sequence are calculated in this paper.

In quasi-synchronous frequency-hopping multiple-access systems where relative delays are restricted within a certain zone, low hit zone frequency-hopping sequences (LHZ FHSs) with favorable partial Hamming correlation properties are desirable. In this paper, we present a new class of LHZ FHS sets with optimal partial Hamming correlation based on t-decimation of m-sequence.

In this paper, a class of additive codes which is referred to as \(\mathbb {Z}_{2}\mathbb {Z}_{2}[u,v]\)-additive codes is introduced. This is a generalization towards another direction of recently introduced \(\mathbb {Z}_{2}\mathbb {Z}_{4}\) codes (Doughterty et al., Appl. Algebra Eng. Commun. Comput. 27(2), 123–138, 7). A MacWilliams-type identity that relates the weight enumerator of a code with

The concept of formal duality was proposed by Cohn, Kumar and Schürmann, which reflects a remarkable symmetry among energy-minimizing periodic configurations. This formal duality was later translated into a purely combinatorial property by Cohn, Kumar, Reiher and Schürmann, where the corresponding combinatorial objects were called formally dual pairs. So far, except the results presented in Li and

Finite frames are sequences of vectors in finite dimensional Hilbert spaces that play a key role in signal processing and coding theory. In this paper, we study the class of tight unit-norm frames for \(\mathbb {C}^{d}\) that also form regular schemes, which we call tight regular schemes (TRS). Many common frames that arise in applications such as equiangular tight frames and mutually unbiased bases

Let m be a positive integer, r ≡ 1 (mod 3) be a prime number, and the order of p modulo rm be \(\frac {\phi (r^{m})}3\), where ϕ is the Euler function. Let \(q=p^{\frac {\phi (r^{m})}3}\) and \(d=\frac {q-1}{r^{m}}\). We investigate the cross correlation distribution between a p-ary m-sequence and its d-decimated sequences. In this paper, we deal with the cases of p = 2 and p = 3. Our results show

In this paper we define a notion of partial APNness and find various characterizations and constructions of classes of functions satisfying this condition. We connect this notion to the known conjecture that APN functions modified at a point cannot remain APN. In the second part of the paper, we find conditions for some transformations not to be partially APN, and in the process, we find classes of

Due to their wide applications in communications, data storage and cryptography, linear codes have received much attention in the past decades. The objective of this paper is to construct a family of linear codes over \({\mathbb F}_{q}\), where q is a prime power. This family of codes has length (qk − 1)t, dimension ek, where k ≥ 2 and e, t are arbitrary integers with 2 ≤ e ≤ t . In some cases, this

Matrices with low coherence have applications in compressed sensing and some other areas. In this paper, we present three deterministic constructions of compressed sensing matrices by using algebraic and combinatorial methods. We show that our results outperform Gaussian random matrices. Moreover, some of our matrices are binary entries, and thus can be used in the embedding operations to get more

Let m1 and m2 be two distinct positive integers with \(d=\gcd (m_{1}, m_{2})\). Let \(\mathbb {F}_{2^{m}}\) be the finite field with 2m elements, where m = m1m2/d. In this paper, we investigate the exponential sums $$ S(a,b)=\sum\limits_{x \in \mathbb{F}_{2^{m}}^{*}}(-1)^{\text{Tr}_{m_{1}}(ax^{\frac {2^{m}-1}{2^{m_{1}}-1}})+\text{Tr}_{m_{2}}(bx^{\frac {2^{m}-1}{2^{m_{2}}-1}})}, $$ where \(a \in \mathbb

The correlation-immune functions serve as an important metric for measuring resistance of a cryptosystem against correlation attacks. Existing literature emphasize on matrices, orthogonal arrays and Walsh-Hadamard spectra to characterize the correlation-immune functions over \(\phantom {\dot {i}\!}\mathbb {F}_{p}\) (p ≥ 2 is a prime). Recently, Wang and Gong investigated the Fourier spectral characterization