Sequences exhibiting zero/low ambiguity zone (ZAZ/LAZ) properties play an important role in contemporary communication and radar systems, particularly in Integrated Sensing and Communication (ISAC), which is an emerging wireless technology by sharing the hardware and bandwidth to perform these two tasks simultaneously. By incorporating nonlinear mapping and parity adaptive factor to design novel exponential functions, this letter proposes three methods for constructing ZAZ/LAZ sequence sets with new parameters. The resulting sequences are cyclically distinct and asymptotically optimal with respect to theoretical bounds.

This paper introduces a novel finite Zak transform (FZT)-aided framework for constructing multiple zero-correlation zone (ZCZ) sequence sets with optimal correlation properties. Specifically, each sequence is perfect with zero auto-correlation sidelobes, each ZCZ sequence set meets the Tang-Fan-Matsufuji bound with equality, and the maximum inter-set cross-correlation of multiple sequence sets meets the Sarwate bound with equality. Our study shows that these sequences can be sparsely expressed in the Zak domain through properly selected index and phase matrices. Particularly, it is found that the maximum inter-set cross-correlation beats the Sarwate bound if every index matrix is a circular Florentine array. Several construction methods of multiple ZCZ sequence sets are proposed, demonstrating both the optimality and high flexibility. Additionally, it is shown that excellent synchronization performance can be achieved by the proposed sequences in orthogonal-time-frequency-space (OTFS) systems.

Nonlinear complexity, as an important measure for assessing the randomness of sequences, is defined as the minimal length of feedback shift registers that can generate a given sequence. This paper establishes the structure ofn-periodic binary sequences with nonlinear complexity larger than or equal to b [3n/4] Based on their structure, an exact enumeration formula for the number of such periodic sequences is determined.

Constant amplitude zero auto-correlation (CAZAC) sequences are widely applied in waveforms for radar and communication systems. Motivated by a recent work [Berggren and Popovi´c, IEEE Trans. Inf. Theory 70(8), 6068-6075 (2024)], this paper advances the approach to generating CAZAC sequences by interleaving Zadoff-Chu (ZC) sequences with permutation polynomials (PPs). We propose one class of high-degree PPs over the integer ring ZN, and utilize them and their inverses to interleave ZC sequences for constructing CAZAC sequences. It is known that a CAZAC sequence can be extended to an equivalence class by five basic operations. We further show that the obtained CAZAC sequences are not covered by the equivalence classes of ZC sequences and interleaved ZC sequences by quadratic PPs and their inverses, and prove the sufficiency of the conjecture by Berggren and Popovi´c in the aforementioned work. In addition, we also evaluate the aperiodic auto-correlation of certain ZC sequences interleaved by quadratic PPs.

Low-rank parity-check (LRPC) codes are the rank-metric analogue of low-density parity-check codes and they found important applications in code-based cryptography. In this paper we investigate a sub-family of LRPC codes, which have a parity-check matrix defined over a subspace${\mathcal {V}}_{\alpha,d}=\langle 1,\alpha, \ldots, \alpha ^{d-1} \rangle _{\mathbb {F}_{q}}\subsetneq \mathbb {F}_{q^{m}} $, where$\mathbb {F}_{q^{m}}$is the finite field of$q^{m}$elements,$\alpha \in \mathbb {F}_{q^{m}}$is an element not in any proper subfield of$\mathbb {F}_{q^{m}}$, and d is a positive integer significantly smaller than m. These codes are termed bounded-degree LRPC (BD-LRPC) codes. BD-LRPC codes are the same as the standard LRPC codes of density 2 when the degree$d=2$, while for degree$d\gt 2$they constitute a proper subset of LRPC codes of density d. Exploiting the structure of${\mathcal {V}}_{\alpha,d}$, the BD-LRPC codes of degree d can uniquely correct errors of rank weight r when$n-k \geq r + u$for certain$u \geq 1$, in contrast to the condition$n-k\geq dr$required for the standard LRPC codes. This underscores the superior decoding capability of the BD-LRPC codes. Moreover, as the code length$n\rightarrow \infty $, when$n/m\rightarrow 0$, the BD-LRPC codes with a code rate of$R=k/n$can be uniquely decodable with radius$\rho =r/n$approaching the Singleton bound$1-R$by letting$\epsilon =u/n\rightarrow 0$; and when$n/m$is a constant, the BD-LRPC codes can have unique decoding radius$\rho = 1-R-\epsilon $for a small$\epsilon $, allowing for$\rho \gt (1-R)/2$with properly chosen parameters. This superior decoding capability is theoretically proved for the case$d=2$and confirmed by experimental results for$d\gt 2$.

Frequency-hopping sequences (FHSs) with low Hamming correlation and wide gaps significantly contribute to the anti-interference performance in FH communication systems. This paper investigates FHSs with optimal Hamming correlation and controlled minimum gaps. We start with the discussion of the upper bounds on the minimum gaps of uniform FHSs and then propose a general construction of optimal uniform wide-gap FHSs with length 2land 3l, which includes the work by Li et al. in IEEE Trans. Inf. Theory, vol. 68, no. 1, 2022 as a special case. Furthermore, we present a recursive construction of FHSs with length 2l, which concatenate shorter sequences of known minimum gaps. It is shown that the resulting FHSs have the same Hamming correlation as the concatenation-ordering sequences. As applications, several known optimal FHSs are used to produce optimal FHSs with controlled minimum gaps.

Quasi-complementary sequence sets (QCSSs) have attracted sustained research interests for simultaneously supporting more active users in multi-carrier code-division multiple-access (MC-CDMA) systems compared to complete complementary codes (CCCs). In this paper, we investigate a novel class of QCSSs composed of multiple CCCs. We derive a new aperiodic correlation lower bound for this type of QCSSs, which is tighter than the existing bounds for QCSSs. We then present a systematic construction of such QCSSs with a flexible alphabet size and a low maximum correlation magnitude, and also show that the constructed aperiodic QCSSs can meet the newly derived bound asymptotically.

Let Fqbe the finite field withqelements andmbe a positive integer. The Fqm-linear low-rank parity-check (LRPC) codes have been used in many cryptographic schemes. Motivated by recent attacks on those schemes, this paper generalizes LRPC codes based on 3-tensors in Fm×m×mq. The generalized LRPC codes are mostly Fq-linear matrix codes, while a particular choice of the 3-tensor is isomorphic to the original Fqm-linear LRPC codes. We first introduce a bilinearT-product over Fmqassociated with a 3-tensorT∈ Fm×m×mq. Based on theT-product, we propose a generic method to expand Fq-linear matrix code from dimensionkto dimensionkmand then use the method to generalize LRPC codes. Finally, we propose two probabilistic polynomial-time decoding algorithms for the generalized LRPC codes under different circumstances. We provide estimates of their decoding failure rates, which, confirmed by experimental results, are almost the same as that of decoding Fqm-linear LRPC codes.

Let n ≥ 3 be an odd integer,d1=3n-1/2-1, d2=3n-2 andube an element of the finite field F3n. This paper shows thatfu(x)=uxd1+xd2is an almost perfect nonlinear (APN) function on F3n if and only if χ(u+1)= χ(u-1)=χ(u), where χ(∙) denotes the quadratic character of F3n. This settles the open problem raised by Ness and Helleseth in IEEE Trans. Inf. Theory 53(7): 2581-2586, 2007, where only the sufficiency part of the result was proved. Furthermore, we investigate the differential spectra offu(x)for elements u satisfying χ(u+1)=χ(u-1) and express them in terms of several quadratic character sums of cubic polynomials.

Nonlinear complexity is an important measure for assessing the randomness of sequences. In this paper we investigate how circular shifts affect the nonlinear complexities of finite-length binary sequences and then reveal a more explicit relation between nonlinear complexities of finite-length binary sequences and their corresponding periodic sequences. Based on the relation, we propose two algorithms that can generate all periodic binary sequences with any prescribed nonlinear complexity.

Due to their weak algebraic structure, low rank parity check (LRPC) codes have been employed in several post-quantum cryptographic schemes. In this paper we propose new improved decoding algorithms for ${\left[ {n,{\text{ }}k} \right]_{{q^m}}}$ LRPC codes of dual rank weight d. The proposed algorithms can efficiently decode LRPC codes with the parameters satisfying n − k = rd − c, where r is the dimension of the error support and c ≤ d − 2. They outperform the original decoding algorithm of LRPC codes when d > 2 and allow for decoding LRPC codes with a higher code rate and smaller values m.

This paper presents a novel class of complex-valued sparse complementary pairs (SCPs), each consisting of a number of zero values and with additional zero-correlation zone (ZCZ) property for the aperiodic autocorrelations and crosscorrelations of the two constituent sequences. Direct constructions of SCPs and their mutually-orthogonal mates based on restricted generalized Boolean functions are proposed. It is shown that such SCPs exist with arbitrary lengths and controllable sparsity levels, making them a disruptive sequence candidate for modern low-complexity, low-latency, and low-storage signal processing applications.

In this work we propose a family of ${\mathbb{F}_q}$-linear lowrank parity check (LRPC) codes based on a bilinear product over $\mathbb{F}_q^m$ defined by a generic 3-tensor over ${\mathbb{F}_q}$. A particular choice of this tensor corresponds to the classical ${\mathbb{F}_{{q^m}}}$-linear LRPC codes; and other tensors yield ${\mathbb{F}_q}$-linear codes, which, with some caveats, can be efficiently decoded with the same idea of decoding LRPC codes. The proposed codes contribute to the diversity of rank metric codes for cryptographic applications, particularly for the cases where attacks utilize ${\mathbb{F}_{{q^m}}}$-linearity to reduce decoding complexity.

In this paper, we present two new families of APN functions. The first family is in bivariate form$\big (x^{3}+xy^{2}+ y^{3}+xy, x^{5}+x^{4}y+y^{5}+xy+x^{2}y^{2} \big)\,\,\vphantom {_{\int _{\int }}}$over${\mathbb F}_{2^{m}}^{2}$. It is obtained by adding certain terms of the form$\sum _{i}(a_{i}x^{2^{i}}y^{2^{i}},b_{i}x^{2^{i}}y^{2^{i}})$to a family of APN functions recently proposed by Gölo&gcaron;lu. The$\vphantom {_{\int _{\int }}}$second family has the form$L(z)^{2^{m}+1}+vz^{2^{m}+1}$over${\mathbb F}_{{2^{3m}}}$, which generalizes a family of APN functions by Bracken et al. from 2011. By calculating the$\Gamma $-rank of the constructed APN functions over${\mathbb F}_{2^{8}}$and${\mathbb F}_{2^{9}}$, we demonstrate that the two families are CCZ-inequivalent to all known families. In addition, the two new families cover two known sporadic APN instances over${\mathbb F}_{2^{8}}$and${\mathbb F}_{2^{9}}$, which were found by Edel and Pott in 2009 and by Beierle and Leander in 2021, respectively.

Let$n$be a positive integer and$p$a prime. The power mapping$x^{p^{n}-3}$over${\mathbb {F}}_{p^{n}}$has desirable differential properties, and its differential spectra for$p=2,\,3$have been determined. In this paper, for any odd prime$p$, by investigating certain quadratic character sums and some equations over${\mathbb {F}}_{p^{n}}$, we determine the differential spectrum of$x^{p^{n}-3}$with a unified approach. The obtained result shows that for any given odd prime$p$, the differential spectrum can be expressed explicitly in terms of$n$. Compared with previous results, a special elliptic curve over${\mathbb {F}}_{p}$plays an important role in our computation for the general case$p \ge 5$.

In this paper we present an interpolation-based decoding algorithm to decode a family of maximum rank distance codes proposed recently by Trombetti and Zhou. We employ the properties of the Dickson matrix associated with a linearized polynomial with a given rank and the modified Berlekamp-Massey algorithm in decoding. When the rank of the error vector attains the unique decoding radius, the problem is converted to solving a quadratic polynomial, which ensures that the proposed decoding algorithm has polynomial-time complexity.

Summary This paper presents a blockchain‐based architecture for our current electronic health record (EHR) systems. Being built on top of existing databases maintained by health providers, the architecture implements a blockchain solution to ensure the integrity of data records and improve interoperability of the systems through tracking all events that happen to the data in the databases. In this proposed architecture, we also introduce a new incentive mechanism for the creation of new blocks on the blockchain. The architecture is independent of any specific blockchain platforms and open to further extensions; hence, it potentially fits in with other electronic record systems that require protection against data misuse.

A new method is used to resolve a long-standing conjecture of Niho concerning the crosscorrelation spectrum of a pair of maximum length linear recursive sequences of length 22m-1 with relative decimation d=2m+2-3, where m is even. The result indicates that there are at most five distinct crosscorrelation values. Equivalently, the result indicates that there are at most five distinct values in the Walsh spectrum of the power permutation f(x)=xdover a finite field of order 22mand at most five distinct nonzero weights in the cyclic code of length 22m-1 with two primitive nonzeros α and αd. The method used to obtain this result proves constraints on the number of roots that certain seventh degree polynomials can have on the unit circle of a finite field. The method also works when m is odd, in which case the associated crosscorrelation and Walsh spectra have at most six distinct values.

In this paper, we apply two-to-one functions over b F2nin two generic constructions of binary linear codes. We consider two-to-one functions in two forms: (1) generalized quadratic functions; and (2) (x2t+x)ewith gcd(t, n)=gcd(e, 2n-1)=1. Based on the study of the Walsh transforms of those functions or their variants, we present many classes of linear codes with few nonzero weights, including one weight, three weights, four weights, and five weights. The weight distributions of the proposed codes with one weight and with three weights are determined. In addition, we discuss the minimum distance of the dual of the constructed codes and show that some of them achieve the sphere packing bound. Moreover, examples show that some codes in this paper have best-known parameters.

In this paper, by the Hasse-Weil bound, we determine the necessary and sufficient condition on coefficients$a_{1},a_{2},a_{3}\in {\mathbb F} _{2^{n}}$with$n=2m$such that$f(x) = {x}^{3\cdot 2^{m}} + a_{1}x^{2^{m+1}+1} + a_{2} x^{2^{m}+2} + a_{3}x^{3}$is an APN function over${\mathbb F}_{2^{n}}$. Our work together with the follow-up work by Chase and Lisoněk indicates that all such APN quadrinomials$f(x)$are affine equivalent to two instances of Gold functions, which resolves the first half of an open problem by Carlet at the International Workshop on the Arithmetic of Finite Fields, 83-107, 2014.

The expansion complexity is a new figure of merit for cryptographic sequences. In this paper, we present an explicit formula of the (irreducible) expansion complexity of ultimately periodic sequences over finite fields. We also provide improved upper and lower bounds on the$N$th irreducible expansion complexity when they are not explicitly determined. In addition, for some infinite sequences with given nonlinear complexity, a tighter upper bound of their$N$th expansion complexity is given.

In this paper, we formulate a generic construction of MRD codes that covers almost all the newly found MRD codes. Among those MRD codes, we particularly investigate the encoding and decoding of a family of nonlinear MRD codes recently by Otal and Özbudak.

This paper presents a blockchain-based architecture for electronic health record (EHR) systems. The architecture is built on top of existing databases maintained by health providers, implements a blockchain solution to improve interoperability of the current EHR systems, prevent tampering and malicious misuse of EHRs by means of tracking all events that happened to the data in the databases. This proposed architecture also introduces a new incentive mechanism for the creation of new blocks in the blockchain. The architecture is independent of any specific blockchain platforms and open to further extensions, hence potentially fits in with other electronic record systems that require protection against tampering and misuse.

The nonlinear complexity of a periodic sequence s is the length of the shortest feedback shift register that can generate s, and its value is upper bounded by the least period of s minus 1. In this paper, a recursive approach that generates all periodic sequences with maximum nonlinear complexity is presented, and the total number of such sequences is determined. The randomness properties of these sequences are also examined.

A Gaussian integer is a complex number whose real and imaginary parts are both integers. This paper proposed a unified construction of perfect Gaussian integer sequences based on cyclic difference sets. It turns out that this construction produces an abundance of perfect Gaussian integer sequences. The proposed construction includes all the sequences recently given by Lee et. al as special cases, and many new families of Gaussian integer sequences. To illustrate, two classes of perfect Gaussian integer sequences defined from Kasami-Welch functions and Helleseth-Gong functions are given.

In this paper, a family of new de Bruijn sequences is proposed through the construction of maximum-length nonlinear feedback shift registers (NFSRs). Let$k$be a positive integer and$p_{0}(x), p_{1}(x), \ldots , p_{k}(x)$be the primitive polynomials in$\mathbb {F}_{2}[x]$with their degrees strictly increasing and pairwise coprime. We determine the cycle structure and adjacency graphs of linear feedback shift registers (LFSRs) with characteristic polynomial$q(x)=\prod \nolimits _{i=0}^{k}p_{i}(x)$. In the case that$p_{0}(x)=1+x$, an algorithm is proposed to produce maximum-length NFSRs from these LFSRs, and it is shown that the algorithm can generate$O(2^{(2^{k}-1)n})~n$-stage maximum-length NFSRs with memory complexity$O(2^{k}kn)$and time complexity$O(2^{n-d_{k}}kn)$, where$n$and$d_{k}$are the degrees of$q(x)$and$p_{k}(x)$, respectively. Finally, we illustrate the proposed algorithm in the case of$k=2$. In this case, we prove that for any integer$n\geq 8$, the algorithm can produce$n$-stage maximum-length NFSRs with time complexity as low as$O(n^{{\rm {log}{log}}(n)}$).

Let m ≥ 3 be an odd integer and p be an odd prime. In this paper, a number of classes of three-weight cyclic codes C(1,e) over Fp, which have parity-check polynomial m1(x)me(x), are presented by examining general conditions on the parameters p, m, and e, where mi(x) is the minimal polynomial of π-i over Fp for a primitive element π of Fpm. Furthermore, for p ≡ 3 (mod 4) and a positive integer e satisfying (pk+ 1) · e ≡ 2 (mod pm - 1) for some positive integer k with gcd(m, k) = 1, the value distributions of the exponential sums T(a, b) = Σx∈FpmωTr(ax+bxe)and S(a, b, c) = Σx∈FpmωTr(ax+bxe+cxs), where s = (pm- 1)/2, are determined. As an application, the value distribution of S(a, b, c) is utilized to derive the weight distribution of the cyclic codes C(1,e,s)with parity-check polynomial m1(x)me(x)ms(x). In the case of p = 3 and even e satisfying the above condition, the dual of the cyclic code C(1,e,s)has optimal minimum distance.

In this paper, the cycle structure and adjacency graphs of a class of linear feedback shift registers (LFSRs) are determined. By recursively applying the D-morphism to the maximum-length LFSRs and representing the cycles by generating functions, a new family of maximum-length nonlinear feedback shift registers (NFSRs) are proposed based on the properties of these LFSRs. The number of NFSRs in the proposed family is also considered.

In this paper, a class of linear feedback shift registers (LFSRs) with characteristic polynomial (1 + x3)p(x) is discussed, where p(x) is a primitive polynomial of degree n > 2. The cycle structure and adjacency graphs of the LFSRs are determined. A new class of de Bruijn sequences is constructed from these LFSRs, and the number of de Bruijn sequences in the class is also considered. To illustrate the efficiency of constructing de Bruijn sequences from these LFSRs, an algorithm for producing some corresponding maximum-length nonlinear feedback shift registers with time and memory complexity O(n) is also proposed.

For an odd prime p and two positive integers m, k such that m/gcd (k,m) ≥ 3 is odd, let d be a positive integer satisfying d(pk+1)=2(mod pm-1). In this paper, the cross-correlation between a p-ary m-sequence and its d-decimated sequences is investigated, and the cross correlation distribution is completely determined. The relationship between the decimations d considered in this paper and some known ones is also studied. This paper generalizes some previous results, and also gives new decimations, which lead to low cross correlation.

In this paper, for an odd prime p and positive integers n, m and e, we present two families of p-ary sequences from decimated Helleseth-Gong sequences and m-sequences and examine their correlation properties. The proposed families of sequences possess low correlation and large linear complexity properties.

This paper studies a class of binomial functions over the finite fields of odd characteristic and characterizes their bentness in terms of the Kloosterman sums. Numerical results show that the proposed class contains bent functions that are affinely inequivalent to all known monomial and binomial ones.