Low Ambiguity Zone (LAZ) sequences are essential in modern integrated sensing and communication (ISAC) systems. Recently, locally perfect nonlinear functions (LPNFs) have been employed to design LAZ sequences with flexible parameters. In this work, we propose three new classes of LPNFs and use them to construct LAZ sequences that offer additional flexible parameters. Notably, all the proposed LAZ sequences are asymptotically optimal with respect to the Ye-Zhou-Fan-Liu-Lei-Tang bounds derived in [IEEE J. Sel. Areas Commun. 40(6), pp. 1809-1822, 2022].

In this paper, we employ general results on the value distributions of perfect nonlinear functions from$\mathbb {F}_{p^{m}}$to$\mathbb {F}_{p}$to give a unified approach to determining the weight distributions of two classes of linear codes over$\mathbb {F}_{p}$constructed from perfect nonlinear functions, where$p$is an odd prime and$m$is an odd number. When$m$is even, we give some mild additional conditions for similar conclusions to hold.

A pure quantum state of$n$parties associated with the Hilbert space$\mathbb {C}^{d_{1}}\otimes \mathbb {C} ^{d_{2}}\otimes \cdots \otimes \mathbb {C} ^{d_{n}}$is called$k$-uniform if all the reductions to$k$-parties are maximally mixed. The$n$partite system is called homogenous if the local dimensions$d_{1}=d_{2}=\cdots =d_{n}$, while it is called heterogeneous if the local dimensions are not all equal.$k$-uniform sates play an important role in quantum information theory. There are much progress in characterizing and constructing$k$-uniform states in homogeneous systems. However, the study of entanglement for heterogeneous systems is much more challenging than that for the homogeneous case. There are very few results known for the$k$-uniform states in heterogeneous systems for$k>3$. We present two general methods to construct$k$-uniform states in the heterogeneous systems for general$k$. The first construction is derived from the error correcting codes by establishing a connection between irredundant mixed orthogonal arrays and error correcting codes. We can produce many new$k$-uniform states such that the local dimension of each subsystem can be a prime power. The second construction is derived from a matrix$H$meeting the condition that$H_{A\times \bar {A}}+H^{T}_{\bar {A}\times A}$has full rank for any row index set$A$of size$k$. These matrix construction can provide more flexible choices for the local dimensions, i.e., the local dimensions can be any integer (not necessarily prime power) subject to some constraints. Our constructions imply that for any positive integer$k$, one can construct$k$-uniform states of a heterogeneous system in many different Hilbert spaces.

Binary${m}$-sequences are those with the largest period$n=2^{m}-1$among the binary sequences produced by linear shift registers with length$m$. They have a wide range of applications in communication since they have several desirable pseudorandom properties, such as balance, uniform pattern distribution, and ideal (classical) autocorrelation. In 1967, Mandelbaum introduced a 2-adic version of classical autocorrelation of binary sequences, called arithmetic autocorrelation, in his research on arithmetic codes. Later, Goresky and Klapper generalized this notion to the nonbinary case and got several properties of arithmetic autocorrelation related to linear shift registers with carry. Recently, Z. Chen et al. showed an upper bound on the arithmetic autocorrelation of binary${m}$-sequences and raised a conjecture on the absolute value distribution of the arithmetic autocorrelation of binary${m}$-sequences. In this paper, we present a general formula for computing arithmetic autocorrelation, from which we completely determine the arithmetic autocorrelation distribution of arbitrary binary${m}$-sequences. In particular, the conjecture raised by Z. Chen et al. is verified.

The cross-correlation between two maximum length sequences ($m$-sequences) of the same period has been studied since the end of 1960s. One open conjecture by Helleseth states that the cross-correlation between any two$p$-ary$m$-sequences takes on the value −1 for at least one shift provided that the decimation$d$obeys$d\equiv 1\,({\mathrm{ mod}}\, p-1)$. This was known as the −1 conjecture. Up to now, the −1 conjecture was confirmed for the following decimations: (1) Niho-type decimations, i.e.,$d=s(p^{n/{2}}-1)+1$, where$s$is an integer; (2) all the complete permutation polynomial (CPP) exponents$d$satisfying$d\equiv 1\, ({\mathrm{ mod}}\, p-1) $; and (3) the additional families of decimations tabulated in this paper. In this paper, we first discuss the connection between the −1 conjecture on cross-correlation of$m$-sequences and CPP exponents, then we confirm the −1 conjecture for a new type of decimations by giving a new class of CPP exponents. The decimations are of the type$d=1+l{(p^{rtm}-1)}/{(r+1)}$over${\mathbb F}_{p^{rtm}}$, where$p$is a prime,$r+1$is an odd prime satisfying$p^{r/{2}} \equiv -1\,({\mathrm{ mod}}\, r+1)$,$t$is an odd integer ($t>2$if$p=2$) with$\gcd (t,r)=1$, and$m$is a positive integer. We transform the problem of determining whether$d$is a CPP exponent into that of investigating the existence of irreducible polynomials over$\mathbb {F}_{p}$with degree$t$satisfying a congruence equation. By a theorem given by Rosen that considered the number of irreducible polynomials with a special congruence relation, we prove that$d$is a CPP exponent over${\mathbb F}_{p^{rtm}}$for sufficiently large$t$. When$m$is odd, our new CPP exponents are of Niho type; thus, we give a new class of CPP exponents of Niho type. When$m$is even, we obtain a new class of CPP exponents which are not of Niho type. As a consequence, we show that the −1 conjecture is true for$d=1+l{(p^{rtm}-1)}/{(r+1)}$when$t$is a sufficiently large integer.

Based on theoretical neuroscience, G. Cotardo and A. Ravagnani (2022) introduced a class of asymmetric binary codes called combinatorial neural codes (CN codes for short), with a “matched metric”$\delta _{r}$called asymmetric discrepancy, instead of the Hamming distance$d_{H}$for usual error-correcting codes. They also presented the Hamming, Singleton and Plotkin bounds for CN codes with respect to$\delta _{r}$and asked how to construct CN codes${\mathcal C}$with large size$| {\mathcal C}|$and minimum$\delta _{r}({\mathcal C})$. In this paper, we first show that a binary code${\mathcal C}$reaches one of the above bounds for$\delta _{r}({\mathcal C})$if and only if${\mathcal C}$reaches the corresponding bounds for$d_{H}$and$r$is sufficiently close to 1. This means that all optimal CN codes come from the usual optimal codes. Then, we present several constructions of CN codes with good and flexible parameters$(n,K, \delta _{r}({\mathcal C}))$by using bent functions.

The purpose of this paper is to determine the 4-adic complexity of the balanced quaternary sequences of period 2(2n−1) with ideal autocorrelation defined by Jang et al. (ISIT, pp. 278-281, 2009). Results show that the 4-adic complexity of such sequences is large enough to resist the attack of the rational approximation algorithm for feedback with carry shift registers.

In this paper, we study the 4-adic complexity of two classes of quaternary sequences of period$2p$with ideal autocorrelation defined by Kim et al. (ISIT, 2009). Our results show that the 4-adic complexity of these two kinds of quaternary sequences is large enough to resist the attack of the rational approximation algorithm.

Via interleaving Ding-Helleseth-Lam sequences, a class of binary sequences with optimal autocorrelation magnitude was constructed (Des. Codes and Cryptogr., 2018). Later, Sun et al. determined the upper and lower bounds of the 2-adic complexity of such sequences (SETA, 2018). In this paper, we determine the exact value of the 2-adic complexity of this class of sequences. The results show that the 2-adic complexity of this class of binary sequences is close to the maximum.

MDS codes and self-dual codes are important families of classical codes in coding theory. Therefore, it is of interest to investigate MDS self-dual codes. The existence of MDS selfdual codes over finite field Fqis completely solved for q is even. In the literature, there are many known constructions of MDS self-dual codes for q is odd. In this paper, we present a unified approach on the existence of MDS self-dual codes with concise statements and simplified proof. It is illustrated that some of known results can also be stated in this framework. Furthermore, we can obtain some new MDS self-dual codes, especially for the case when q is not a square.

We determine the 2-adic complexity of the Ding-Helleseth-Martinsen (DHM) binary sequences by using cyclotomic numbers of order four, “Gauss periods” and “quadratic Gauss sums” on finite field Fq and valued in Z2N-1, where q ≡ 5 (mod 8) is a prime number and N = 2q is the period of the DHM sequences.

This paper contributes to increase our knowledge on generalized bent functions (including generalized bent Boolean functions and generalized $p$ -ary bent functions with odd prime $p$ ) by bringing new results on their characterization and construction in arbitrary characteristic. More specifically, we first investigate relations between generalized bent functions and bent functions by the decomposition of generalized bent functions. This enables us to completely characterize generalized bent functions and $\mathbb Z_{p^{k}}$ -bent functions by some affine space associated with the generalized bent functions. We also present the relationship between generalized bent Boolean functions with an odd number of variables and generalized bent Boolean functions with an even number of variables. Based on the well-known Maiorana-McFarland class of Boolean functions, we present some infinite classes of generalized bent Boolean functions. In addition, we introduce a class of generalized hyperbent functions that can be seen as generalized Dillon's $PS$ functions. Finally, we solve an open problem related to the description of the dual function of a weakly regular generalized bent Boolean function with an odd number of variables via the Walsh-Hadamard transform of their component functions, and we generalize these results to the case of odd prime.

A pure quantum state is called k-uniform if all its reductions to k-qudit are maximally mixed. We investigate the general constructions of k-uniform pure quantum states of n subsystems with d levels. We provide one construction via symmetric matrices and the second one through the classical error-correcting codes. There are three main results arising from our constructions. First, we show that for any given even n ≥ 2, there always exists an n/2-uniform n-qudit quantum state of level p for sufficiently large prime p. Second, both constructions show that there exist k-uniform n-qudit pure quantum states such that k is proportional to n, i.e., k = Ω(n) although the construction from symmetric matrices in general outperforms the one by error-correcting codes. Third, our symmetric matrix construction provides a positive answer to the open question on whether there exists a 3-uniform n-qudit pure quantum state for all n ≥ 8. In fact, we can further prove that, for every k, there exists a constant Mksuch that there exists a k-uniform n-qudit quantum state for all n ≥ Mk. In addition, by using the concatenation of algebraic geometry codes, we give an explicit construction of k-uniform quantum state when k tends to infinity.

In this paper, we investigate properties of generalized bent Boolean functions and 2k-bent (i.e., negabent, octabent, hexadecabent, et al.) Boolean functions in a uniform framework. From the Hadamard matrices, Hodzic and Pasalic presented sufficient conditions for generalized bent functions. Using cyclotomic fields and the decomposition of generalized bent functions, we generalize their results, prove that Hodzic and Pasalic's conditions of generalized bent functions are not only sufficient but also necessary, and completely characterize generalized bent functions in terms of their component functions. Furthermore, we present a secondary construction of bent functions or semibent functions from generalized bent functions. Finally, we give the relations of generalized bent functions and 2k-bent functions, demonstrate that 2k-bent functions are actually a special class of generalized bent functions, and completely characterize 2k-bent functions.

In this paper, we present a construction of linear codes over F2tfrom Boolean functions, which is a generalization of Ding's method. Based on this construction, we give two classes of linear codes C̃fand Cfover F2tfrom a Boolean function f : Fq→ F2, where q = 2nand F2tis some subfield of Fq. The complete weight enumerator of C̃fcan be easily determined from the Walsh spectrum of f , while the weight distribution of the code Cfcan also be easily settled. Particularly, the number of nonzero weights of C̃fand C f is the same as the number of distinct Walsh values of f. As applications of this construction, we show several series of linear codes over F2twith two or three weights by using bent, semibent, monomial and quadratic Boolean function f.

The generalized Hamming weights dr(C) of a linear code C are a natural generalization of the minimum Hamming distance d(C)[=d1(C)] and have become an important research object in coding theory since Wei's originary work in 1991. In this paper, two general formulas on d(C) for irreducible cyclic codes are presented using Gauss sums and the weight hierarchy {d1(C), d2(C), ... , dk(C)} (k= dim C) is completely determined for several cases.

In this paper, the value of Imax(C) for codebooks C = C(D) constructed by certain almost difference sets D in Fqxis determined and expressed in terms of Jacobi sums from which it shows that such codebooks nearly meet the Welch bound. This result is an answer of a question raised by C.Ding and T.Feng in [2]. We also present another series of codebooks which nearly meet the Welch bound, where the codebook is constructed by a subset R = Fq1⊕ Fq2. When q1= q2, Dεis an almost difference set of R.

The stabilizer method for constructing a class of asymmetric quantum codes (AQC), called additive AQC, has been established by Aly et.al. In this paper, we present a new characterization of AQC, which generalizes a result of the symmetric case known previously. As an application of the characterization, we establish a relationship of AQC with classical error-correcting codes and show a few examples of good AQC with specific parameters. By using this relationship, we obtain an asymptotic bound on AQCs from algebraic geometry codes.

Symmetric Boolean functions with even variables2kand maximum algebraic immunity AI(f)=khave been constructed in Braeken's thesis (2006). In this paper, we show more constructions of such Boolean functions including the generalization of a result and prove a conjecture raised in Braeken's thesis (2006).

Let q=pmwhere p is an odd prime, mges2, and 1lesklesm-1. Let Tr be the trace mapping from Fqto Fpand zetap=e2pii/pbe a primitive pth root of unity. In this paper, we determine the value distribution of the following exponential sums: SigmaxisinFqchi(alphaxpk+1+betax2) (alpha, betaisinFq) where chi(x)=zetapTr(x)is the canonical additive character of Fq. As applications, we have the following. 1) We determine the weight distribution of the cyclic codes C1and C2over Fpt with parity-check polynomial h2(x)h3(x) and h1(x)h2(x)h3(x), respectively, where t is a divisor of d=gcd(m, k), and h1(x), h2(x) , and h3(x) are the minimal polynomials of pi-1, pi-2, and pi-(pk+1)over Fpt, respectively, for a primitive element pi of Fq. 2) We determine the correlation distribution between two m-sequences of period q-1. Moreover, we find a new class of p-ary bent functions. This paper extends the results in Feng and Luo (2008).

In this paper, we will study the exponential sum SigmaxisinFqx(alphax(pk+1)/2+betax) that is related to the generalized Coulter-Matthews function x(pk+1)/2 with k/gcd. As applications, we obtain the following: the correlation distribution of a p-ary m-sequence and a decimated m-sequence of degree pk+1/2; the weight distribution of the cyclic code whose dual has two zeros pi-1and pi-((pk+1)/2).

In this paper we present a unified way to determine the values and their multiplicities of the exponential sums$$\sum _{x\in\BBF _{q}}\zeta _{p}^{{\rm Tr}\left(af(x)+bx\right)}\left(a,b\in \BBF _{q},q=p^{m},p\ge 3\right)$$for all perfect nonlinear functions$f$which is a Dembowski–Ostrom polynomial or$p\!=\!3$,$f\!=\!x^{{ 3^{k} + 1}\over { 2}}$where$k$is odd and$(k,m)\!=\!1.\break$As applications, we determine 1) the correlation distribution of the$m$-sequence$\left \{a_{\lambda }= {\rm Tr}(\gamma ^{\lambda })\right \}({\lambda =0,1,\ldots })$and the sequence$\left \{b_{\lambda }= {\rm Tr}\left (f(\gamma ^{\lambda })\right)\right \}({\lambda =0,1,\ldots })$over$\BBF _{p}$where$\gamma $is a primitive element of$\BBF _{q}$and 2) the weight distributions of the linear codes over$\BBF _{p}$defined by$f$.

In this note, it is proved that for each odd positive integer n there are exactly two n-variable symmetric Boolean functions with maximum algebraic immunity.

We generalize a characterization of p-ary (p is a prime) quantum codes given by Feng and Xing to q-ary (q is a prime power) quantum codes. This characterization makes it possible to convert an asymptotic bound of Stichtenoth and Xing for nonlinear algebraic geometry codes to a quantum asymptotic bound. Besides, we also investigate the asymptotic behavior of quantum codes

A finite Gilbert-Varshamov (GV) bound for pure stabilizer (binary and nonbinary) quantum error correcting codes is presented in analogy to the GV bound for classical codes by using several enumerative results in finite unitary geometry. From this quantum GV bound we obtain several new binary quantum codes in a nonconstructive way having better parameters than the known codes.

Several new results on the nonexistence of generalized bent functions are proved by using properties of the decomposition law of primes and the class group of imaginary Abelian number fields.

We prove that nonbinary quantum stabilizer codes with parameters [[n, k, d]]/sub p/ = [[6, 2, 3]]/sub p/ and [[7, 3, 3]]/sub p/ exist for all odd primes p by using graph machinery, given by Schlingemann and Werner(see Phys. Rev. A, vol.65, no.012308, 2001), with a little number theory and combinatorics.

We give direct and recursive constructions for aperiodic and periodic complementary sequences. Using these constructions, many missing entries in the table of Bomer and Antweiler (1990) can be filled.