A uniform linear array (ULA) represents a fundamental sparse array configuration characterized by constant inter-element spacing. In this letter, for any positive integer$r$, we propose an$r$th-order generalized five-ULA sparse array ($r$-G5USA), which includes super augmented nested array (SANA) and a class of enhanced multi-ULA sparse array (EMUSA) structure as special cases. Closed-form expressions are derived for both the array configuration and the weight function of$r$-G5USA. Specifically, the proposed$r$-G5USA not only maintains a high number of uniform degrees of freedom (uDOFs) on the order of$\mathcal {O}(\frac{2N^{2}}{3} + \frac{2N}{3})$, but also enables flexible control of the first$2r-1$or$2r-2$weight functions, reducing their values to unity. Simulation results demonstrate that$r$-G5USA exhibits superior performance over existing sparse arrays in terms of uDOFs, coupling leakage (CL), and direction-of-arrival (DOA) estimation accuracy.

Fractional repetition codes (FRCs)/Generalized FRCs (GFRCs) are classes of minimum bandwidth regenerating codes that play a key role in distributed storage systems. In this paper, first, we explore the inherent connection between biregular bipartite graphs and FRCs/GFRCs. Using the structural characteristics of these graphs, we determine the lower bound of the file size of the FRCs/GFRCs related to these graphs. Specifically, we leverage the eigenvalues of biregular bipartite graphs and derive several lower bounds on the file size of FRCs/GFRCs with flexible parameters and reconstruction degrees.

Frequency hopping sequences (FHSs) play a crucial role in frequency hopping (FH) communication systems due to their strong anti-interference ability, low interception probability, high confidentiality and strong concealment. The objective of this paper is to construct FHSs for quasi-synchronous frequency-hopping multiple access (FHMA) communication systems that simultaneously achieve optimal no-hit zone (NHZ) length and optimal gap. To accomplish this, the paper first derives tighter upper bounds for the gap size in both periodic and aperiodic scenarios under the assumption that all frequencies within the designated frequency slot set are fully utilized. Subsequently, this paper proposes a class of wide-gap frequency hopping sequences (WGFHSs) and a class of multi-timeslot wide-gap frequency hopping sequences (MTWGFHSs), both of which simultaneously exhibit optimal NHZ length and optimal gap.

Subfield codes of linear codes over finite fields have recently received much attention since they can produce optimal codes, which may have applications in secret sharing, authentication codes and association schemes. In this paper, we first present a construction framework of 3-dimensional linear codesCf,gover Fqmparameterized by any two functionsf,gover Fqm, and then study the properties of six types ofCf,g, its punctured codeC*f,gand their corresponding subfield codes over Fq. The classification ofCf,gis based on special choices off,gas trace function, norm function, almost bent function, Boolean bent function or a combination of these functions. For the first two types ofCf,g, we explicitly determine the weight distributions and dualities ofCf,g, C*f,gand their subfield codes over Fq. The remaining four types ofCf,gare restricted toq= 2, and the weight distributions and dualities of the subfields codeC(q)f,gandC*f,g(q)are completely determined. Most of the resultant linear codes (over Fqmor over Fq) have few weights. Some of them are optimal and some have the best-known parameters according to the tables maintained at http://www.codetables.de. In fact, 16 infinite families of optimal linear codes are produced in this paper. As a byproduct, a family of [24m-2, 2m+ 1, 24m-3] quaternary Hermitian self-orthogonal codes are obtained withm≥ 2. As an application, we present several infinite families of 2-designs or 3-designs with some of the codes presented in this paper.

The defect of an$[n,k,d]$linear code is defined as$s({\mathcal{ C}})=n-k+1-d$. Codes with$s({\mathcal{ C}})=0$are called maximum distance separable (MDS), while codes with$s({\mathcal{ C}})=s({\mathcal{ C}}^{\perp})=1$are called near maximum distance separable (NMDS). NMDS codes correspond to interesting objects in finite geometry and have nice applications in combinatorics and cryptography. There have been many constructions of NMDS codes, but most of them are focus on fixed$q$or$k$, except for constructions from elliptic curves. Roth and Lempel (IEEE Trans. Inf. Theory 1989) constructed a type of linear codes (referred as Roth-Lempel codes), and presented the necessary and sufficient conditions of Roth-Lempel code to be MDS. Especially, they pointed out that the resultant MDS codes is not linearly equivalent to Reed-Solomn codes. In this paper, the NMDS properties of Roth-Lempel codes will be analyzed. We also obtain the necessary and sufficient condition of Roth-Lempel codes to be NMDS, and further completely determine the weight distributions of Roth-Lempel codes with length$q+2$and dimension$3\leq k\leq q$. Besides, by analyzing the upper bound for the code lengths of elliptic curve MDS codes, we illustrate the linearly inequivalence of Roth-Lempel NMDS codes and elliptic curve NMDS codes when their corresponding code lengths exceed$4(q+2\sqrt {q}+1)/5+1$.

Cross Z-complementary pairs (CZCPs) are a special kind of Z-complementary pairs (ZCPs) having zero autocorrelation sums around the in-phase and end-shift positions and zero cross-correlation sums around the end-shift positions. CZCPs can be crucial in designing optimal training sequences for broadband spatial modulation (SM) systems over frequency-selective channels. In this paper, we focus on designing new CZCPs with large cross Z-complementary ratio (CZCR). A construction framework of CZCPs with a large ZCZ ratio is proposed using Turyn’s method on some seed CZCPs and GCPs. By choosing suitably the seed CZCPs, we obtain 24 classes of new CZCPs with large CZCR. Especially, if the GCP is strengthened, our resultant CZCPs have the maximum$\mathrm {\mathbf{CZCR}}\approx \frac {M-1}{M}$for$M\in \{6,12,24,28,48,56\}$. We also obtain optimal CZCPs with new parameters (28, 13), (48, 23), (56, 27), (96, 47) and (112, 55), which can be extended to$(96N,47N)$-CZCPs and$(112N,55N)$-CZCPs respectively for any Golay number$N$.

We study the Singleton-type bound that provides an upper limit on the minimum distance of locally repairable codes. We present an improved bound by carefully analyzing the combinatorial structure of the repair sets. Thus, we show the previous bound is unachievable for certain parameters. We then also provide explicit constructions of optimal codes which show that for certain parameters the new bound is sharp. Additionally, as a byproduct, some previously known codes are shown to attain the new bound and are thus proved to be optimal.

In frequency hopping (FH) communication systems, frequency hopping sequences (FHSs) are crucial in determining the system’s anti-jamming performance. If FHSs can ensure a wide-gap between two adjacent frequency points to avoid the frequency points with high interference probability, it will significantly improve the FH communication system’s anti-interference ability. Moreover, if each frequency point appears at the same number of times in a sequence period, the system’s anti-electromagnetic interference will be enhanced. Therefore, it is desirable to employ FHSs with low Hamming autocorrelation, wide frequency-hopping gap, and good uniformity in practical applications. However, to the best of our knowledge, no such infinite classes of FHSs have been reported in the literature to date. This paper aims to present two constructions of uniform wide-gap frequency-hopping sequences (WGFHSs) by concatenating two or three adequately designed sequences. For the first time, we obtain two infinite classes of WGFHSs, which are optimal with respect to the well-known Lempel-Greenberger bound.

The Singleton-type bound that provides an upper limit on the minimum distance of locally repairable codes is studied. An improved bound is presented by carefully analyzing the combinatorial structure of the repair sets. Thus, we show the previous bound is unachievable for certain parameters. Additionally, as a byproduct, some previously known codes are shown to attain the new bound and are thus proved to be optimal.

Locally repairable codes are desirable for distributed storage systems to improve the repair efficiency. In this paper, a connection between locally repairable codes with multiple disjoint repair sets and packings is derived under the condition that each repair set contains exactly one check symbol. Particularly, conditions under which an optimal locally repairable code corresponds to a packing are also characterized. As an application of this connection, some optimal locally repairable codes can be obtained by packings. Specifically, two constructions of locally repairable codes are proposed which not only generalize some known explicit constructions but also give optimal locally repairable codes with flexible new parameters.

As a class of optimal combinatorial objects, bent functions have important applications in cryptography, sequence design, and coding theory. Bent idempotents are a subclass of bent functions and of great interest, since they can be stored in less space and allow faster computation of the Walsh-Hadamard transform. The objective of this paper is to present a generic construction of bent functions from known ones. It includes the previous constructions of bent functions by Mesnager and Xu et al. as special cases, and produces new bent functions, which cannot be produced by earlier ones. In particular, it also generates infinite families of bent idempotents over F22mof any algebraic degree between 2 and m. This together with a recent construction by Su and Tang gives a positive answer to an open problem on bent idempotents proposed by Carlet. In addition, an infinite family of anti-self-dual bent functions is obtained in which the sum of any three distinct functions is again an anti-self-dual bent function in this family. This solves an open problem recently proposed by Mesnager.

Frequency-hopping sequences (FHSs) with favorable partial Hamming correlation properties have important applications in many synchronization and multiple-access systems. Strictly optimal FHSs are those FHSs with optimal partial Hamming autocorrelation irrespective of the correlation window length. In this paper, strictly optimal FHSs are investigated from a combinatorial approach. A generic connection between strictly optimal FHSs and disjoint cyclic perfect Mendelsohn difference families is established. By virtue of this connection, new strictly optimal FHSs are generated from some disjoint CPMDFs. These strictly optimal FHSs have new parameters not covered in the literature.

The theory of cyclotomy dates back to Gauss and has a number of applications in combinatorics, coding theory, and cryptography. Cyclotomy over a residue class ring${\BBZ}_{v}$can be divided into classical cyclotomy or generalized cyclotomy, depending on$v$prime or composite. In this paper, we introduce a generalized cyclotomy of order$d$over${\BBZ}_{p_{1}^{e_{1}}p_{2}^{e_{2}},\ldots, p_{n}^{e_{n}}}$, which includes Whiteman's and Ding-Helleseth's generalized cyclotomy as special cases. Here,$p_{1},p_{2},\ldots,p_{n}$are pairwise distinct odd primes satisfying$d\vert (p_{i}-1)$for all$1\leq i\leq n$and$e_{1},e_{2},\ldots,e_{n}$are positive integers. We derive some basic properties of the corresponding cyclotomic numbers and obtain a general formula to compute them via classical cyclotomic numbers. As applications, we completely solve an open problem and a conjecture on Whiteman's generalized cyclotomy of order four over${\BBZ}_{p_{1}p_{2}}$. Besides, we also construct an infinite series of near-optimal codebooks over${\BBZ}_{p_{1}p_{2}}$, as well as some infinite series of asymptotically optimal difference systems of sets over${\BBZ}_{p_{1}^{e_{1}}p_{2}^{e_{2}},\ldots,p_{n}^{e_{n}}}$.