Let v=p1m1p2m2…ptmt be the canonical prime factorization of v. In this paper, we give a construction of optimal ((s+1)×v,s+1,1) two-dimensional optical orthogonal codes with both at most one-pulse per wavelength and at most one-pulse per time slot, where s | gcd(p1-1,p2-1,...,pt-1). The method is much simpler than that in [1]. Optimal (m×v,k,1) two-dimensional optical orthogonal codes are also constructed based on the Steiner system S[2,k,m].

In this letter, motivated by the research of Tian et al., two constructions of asymptotically optimal codebooks in regard to the Welch bound with additive and multiplicative characters are provided. The parameters of constructed codebooks are new, which are different from those in the letter of Tian et al.

In this paper, we consider the Reed-Solomon codes over Fqm with evaluations in a subfield Fq. By the “virtual extension”, we can embed these codes into homogeneous interleaved Reed-Solomon codes. Based on this property and the collaborative decoding algorithm, a new probabilistic decoding algorithm that can correct errors up to $rac{m}{m+1}(n-k)$ for these codes is proposed. We show that whether the new decoding algorithm fails or not is only dependent on the error. We also give an upper bound on the failure probability of the new decoding algorithm for the case s=2. The new decoding algorithm has some advantages over some known decoding algorithms.

Linear codes with locality r and availability t have a wide application in distribution storage because they permit local repair and parallel accesses of hot data. In this letter, the locality and availability of some linear codes based on finite geometry are given. According to these results, we give some linear codes that have higher rate than known codes with the same locality and availability.

Sidon space is an important tool for constructing cyclic subspace codes. In this letter, we construct some Sidon spaces by using primitive elements and the roots of some irreducible polynomials over finite fields. Let q be a prime power, k, m, n be three positive integers and $ ho= lceil rac{m}{2k} ceil-1$, $ heta= lceil rac{n}{2m} ceil-1$. Based on these Sidon spaces and the union of some Sidon spaces, new cyclic subspace codes with size $rac{3(q^{n}-1)}{q-1}$ and $rac{ heta ho q^{k}(q^{n}-1)}{q-1}$ are obtained. The size of these codes is lager compared to the known constructions from [14] and [10].

For an [n, k, d] (r, δ)-locally repairable codes ((r, δ)-LRCs), its minimum distance d satisfies the Singleton-like bound. The construction of optimal (r, δ)-LRC, attaining this Singleton-like bound, is an important research problem in recent years for thier applications in distributed storage systems. In this letter, we use Reed-Solomon codes to construct two classes of optimal (r, δ)-LRCs. The optimal LRCs are given by the evaluations of multiple polynomials of degree at most r - 1 at some points in Fq. The first class gives the [(r + δ - 1)t, rt - s, δ + s] optimal (r, δ)-LRC over Fq provided that r + δ + s - 1≤q, s≤δ, s

In the letter, two classes of optimal codebooks and asymptotically optimal codebooks in regard to the Levenshtein bound are presented, which are based on mutually unbiased bases (MUB) and approximately mutually unbiased bases (AMUB), respectively.

Mutually unbiased bases (MUBs) are widely used in quantum information processing and play an important role in quantum cryptography, quantum state tomography and communications. It’s difficult to construct MUBs and remains unknown whether complete MUBs exist for any non prime power. Therefore, researchers have proposed the solution to construct approximately mutually unbiased bases (AMUBs) by weakening the inner product conditions. This paper constructs q AMUBs of ℂq, (q + 1) AMUBs of ℂq-1 and q AMUBs of ℂq-1 by using character sums over Galois rings and finite fields, where q is a power of a prime. The first construction of q AMUBs of ℂq is new which illustrates K AMUBs of ℂK can be achieved. The second and third constructions in this paper include the partial results about AMUBs constructed by W. Wang et al. in [9].

Optical orthogonal signature pattern codes (OOSPCs) have attracted great attention due to their important application in the spatial code-division multiple-access network for image transmission. In this paper, we give a construction for OOSPCs based on cyclic codes over Fp. Applying this construction with the Reed-Solomon codes and the generalized Berlekamp-Justesen codes, we obtain two classes of asymptotically optimal OOSPCs.

Permutation polynomials over finite fields have been widely studied due to their important applications in mathematics and cryptography. In recent years, 2-to-1 mappings over finite fields were proposed to build almost perfect nonlinear functions, bent functions, and the semi-bent functions. In this paper, we generalize the 2-to-1 mappings to m-to-1 mappings, including their construction methods. Some applications of m-to-1 mappings are also discussed.