Let R=Z4 be the integer ring mod 4 and C be a linear code over R. The code C is called a triple cyclic code of length (r, s, t) over R if the set of its coordinates can be partitioned into three parts so that any cyclic shift of the coordinates of the three parts leaves the code invariant. These codes can be viewed as R[x]-submodules of R[x]/×R[x]/×R[x]/. In this paper, we determine the generator polynomials and the minimum generating sets of this kind of codes.

In this letter, as a generalization of Luo et al.'s constructions, a construction of codebook, which meets the Welch bound asymptotically, is proposed. The parameters of codebook presented in this paper are new in some cases.

As an optimal combinatorial object, cyclic perfect Mendelsohn difference family (CPMDF) was introduced by Fuji-Hara and Miao to construct optimal optical orthogonal codes. In this paper, we propose a direct construction of disjoint CPMDFs from the Zeng-Cai-Tang-Yang cyclotomy. Compared with a recent work of Fan, Cai, and Tang, our construction doesn't need to depend on a cyclic difference matrix. Furthermore, strictly optimal frequency-hopping sequences (FHSs) are a kind of optimal FHSs which has optimal Hamming auto-correlation for any correlation window. As an application of our disjoint CPMDFs, we present more flexible combinatorial constructions of strictly optimal FHSs, which interpret the previous construction proposed by Cai, Zhou, Yang, and Tang.

Hybrid analog/digital precoding has attracted growing attention for millimeter wave (mmWave) communications, since it can support multi-stream data transmission with limited hardware cost. A main challenge in implementing hybrid precoding is that the channels will exhibit frequency-selective fading due to the large bandwidth. To this end, we propose a practical hybrid precoding scheme with finite-resolution phase shifters by leveraging the correlation among the subchannels. Furthermore, we utilize the sparse feature of the mmWave channels to design a low-complexity algorithm to realize the proposed hybrid precoding, which can avoid the complication of the high-dimensionality eigenvalue decomposition. Simulation results show that the proposed hybrid precoding can approach the performance of unconstrained fully-digital precoding but with low hardware cost and computational complexity.

For any positive integer n, define an iterated function $f(n)=left{ egin{array}{ll} n/2, & mbox{ $n$ even, } 3n+1, & mbox{ $n$ odd. } end{array} ight.$ Suppose k (if it exists) is the lowest number such that fk(n)

In this short correspondence, (1+uv)-constacyclic codes over the finite non-chain ring R[v]/(v2+v) are investigated, where R=F2+uF2 with u2=0. Some structural properties of this class of constacyclic codes are studied. Further, some optimal binary linear codes are obtained from these constacyclic codes.

Let $mathbb{F}_q$ be a finite field of q elements, $R=mathbb{F}_q+umathbb{F}_q$ (u2=0) and D2n= be a dihedral group of order n. Left ideals of the group ring R[D2n] are called left dihedral codes over R of length 2n, and abbreviated as left D2n-codes over R. Let n be a positive factor of qe+1 for some positive integer e. In this paper, any left D2n-code over R is uniquely decomposed into a direct sum of concatenated codes with inner codes Ai and outer codes Ci, where Ai is a cyclic code over R of length n and Ci is a linear code of length 2 over a Galois extension ring of R. More precisely, a generator matrix for each outer code Ci is given. Moreover, a formula to count the number of these codes is obtained, the dual code for each left D2n-code is determined and all self-dual left D2n-codes over R are presented, respectively.

Generalized quasi-cyclic (GQC) codes with arbitrary lengths over the ring $mathbb{F}_{q}+umathbb{F}_{q}$, where u2=0, q=pn, n a positive integer and p a prime number, are investigated. By the Chinese Remainder Theorem, structural properties and the decomposition of GQC codes are given. For 1-generator GQC codes, minimum generating sets and lower bounds on the minimum distance are given.

Let Fq be a finite field of cardinality q, R=Fq[u]/=Fq+uFq+u2Fq+u3Fq (u4=0) which is a finite chain ring, and n be a positive integer satisfying gcd(q,n)=1. For any $delta,alphain mathbb{F}_{q}^{ imes}$, an explicit representation for all distinct (δ+αu2)-constacyclic codes over R of length n is given, and the dual code for each of these codes is determined. For the case of q=2m and δ=1, all self-dual (1+αu2)-constacyclic codes over R of odd length n are provided.

In this paper, new classes of binary generalized cyclotomic sequences of period 2pm+1qn+1 are constructed. These sequences are balanced. We calculate the linear complexity of the constructed sequences with a simple method. The results show that the linear complexity of such sequences attains the maximum.

As a generalization of perfect nonlinear (PN) functions, zero-difference balanced (ZDB) functions play an important role in coding theory, cryptography and communications engineering. Inspired by a foregoing work of Liu et al. [1], we present a class of ZDB functions with new parameters based on the cyclotomy in finite fields. Employing these ZDB functions, we obtain simultaneously optimal constant composition codes and perfect difference systems of sets.

Feedback with carry shift registers (FCSRs) implemented using Galois representation have been found to have a weakness called LFSRization. It leads to powerful attacks against the stream ciphers based on them. A new representation called ring representation has been proposed to avoid the attacks. It was considered to circumvent the weaknesses of Galois FCSRs. This correspondence presents a class of ring FCSRs, which meet the implementation criteria, but are still possible to maintain linear behavior for several clock cycles. Their LFSRization probability and how to improve their security are also mentioned.

In this paper, we study self-dual cyclic codes of length n over the ring R=Z4[u]/, where n is an odd positive integer. We define a new Gray map φ from R to Z42. It is a bijective map and maintains the self-duality. Furthermore, we give the structures of the generators of cyclic codes and self-dual cyclic codes of odd length n over the ring R. As an application, some self-dual codes of length 2n over Z4 are obtained.

Compressed sensing theory provides a new approach to acquire data as a sampling technique and makes sure that a sparse signal can be reconstructed from few measurements. The construction of compressed sensing matrices is a main problem in compressed sensing theory (CS). In this paper, the deterministic constructions of compressed sensing matrices based on affine singular linear space over finite fields are presented and a comparison is made with the compressed sensing matrices constructed by DeVore based on polynomials over finite fields. By choosing appropriate parameters, our sparse compressed sensing matrices are superior to the DeVore's matrices. Then we use a new formulation of support recovery to recover the support sets of signals with sparsity no more than k on account of binary compressed sensing matrices satisfying disjunct and inclusive properties.

In this letter, we first investigate some new properties of a known power residue frequency-hopping sequence (FHS) set which is established as an optimal one-coincidence frequency-hopping sequence (OC-FHS) set with near-optimal set size. Next, combining the mathematical structure of power residue theory with interleaving technique, we present a new class of optimal OC-FHS set, using the Chinese Remainder Theorem (CRT). As a result, one optimal OC-FHS set with prime length is extended to another optimal OC-FHS set with composite length in which the construction preserves the maximum Hamming correlation (MHC) and the set size as well as the optimality of the Lempel-Greenberger bound.

Many linear codes with two or three weights have recently been constructed due to their applications in consumer electronics, communication, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, two classes of p-ary linear codes with two or three weights are presented. The first class of linear codes with two or three weights is obtained from a certain non-quadratic function. The second class of linear codes with two weights is obtained from the images of a certain function on $mathbb{F}_{p^m}$. In some cases, the resulted linear codes are optimal in the sense that they meet the Griesmer bound.