A (1,k)-overlap-free code, motivated by applications in DNA-based data storage systems and synchronization between communication devices, is a set of words in which no prefix of lengthtof any word is the suffix of any word for every integer t such that 1 ≤t≤k. A (1,n— 1)-overlap-free code of lengthnis said to be non-overlapping. We provide a construction forq-ary (1,k)-overlap-free codes of length 2k, which can be viewed as a generalization of the Zero Block Construction presented by Blackburn, Esfahani, Kreher and Stinson recently over a binary alphabet, and analyze the asymptotic behavior of their sizes. Whenn≥ 2k, an explicit general lower bound and an asymptotic lower bound for the size of an optimalq-ary (1,k)-overlap-free code of lengthnare presented. The exact value of the maximum size ofq-ary (1, 2)-overlap-free codes of lengthnis determined for anyn≥ 4, and a construction forq-ary (1,k)-overlap-free codes of lengthk+ 2 is given.

Optical orthogonal codes have applications in optical code-division multiple access communication systems. They can also be used to construct protocol sequences for multiuser collision channel without feedback, and constant weight codes for error detection and correction. Direct constructions with explicit codewords are presented to settle the existence of a J-optimal$(v,4,1)$-optical orthogonal code with$\lfloor (v-1)/12\rfloor $codewords for any positive integer$v\neq 25$. As a corollary, it is shown that an optimal$(v,6,4)$-cyclically permutable constant weight code with$\lfloor (v-1)/12\rfloor $codewords exists for any positive integer$v\neq 25$.

A multichannel conflict-avoiding code (MC-CAC) is a collection of two-dimensional codewords represented by zero-one matrices, and any pair of distinct codewords have at most one overlapping 1 regardless of the relative time offsets. Such codes are of practical interest as they are able to provide a hard guarantee that each active user has a successful transmission within every consecutive$n$time slots in a wireless network with multiple asynchronous collision channels. In a more practical setting, it is assumed that in each time slot each source node can only pick one channel and send one packet in the chosen channel, and hence the at most one-packet per time slot (AM-OPPTS) restriction is appended to an MC-CAC. Only upper bounds for the number of codewords of an AM-OPPTS MC-CAC with weights three and four were known in the literature, and constructions for AM-OPPTS MC-CACs have not been explored and analyzed systematically. This paper is devoted to establishing combinatorial constructions for AM-OPPTS MC-CACs by introducing holey group divisible packings with prescribed automorphism groups. As applications of our constructions, the exact values of the sizes of both an optimal AM-OPPTS MC-CAC$(m,n,3)$and an optimal AM-OPPTS MC-CAC$\left({m,p^{r},\frac {p+1}{2}}\right)$are determined for certain$m,n,p$and$r$.

Constant dimension codes (CDCs), as special subspace codes, have received a lot of attention due to their application in random network coding. This paper introduces a family of new codes, called rank metric codes with given ranks (GRMCs), to generalize the parallel construction in [Xu and Chen, IEEE Trans. Inf. Theory, 64 (2018), 6315-6319] and the classic multilevel construction. A Singleton-like upper bound and a lower bound for GRMCs derived from Gabidulin codes are given. Via GRMCs, two effective constructions for CDCs are presented by combining the parallel construction and the multilevel construction. Many CDCs with larger size than the previously best known codes are given. The ratio between the new lower bound and the known upper bound for (4δ, 2δ, 2δ)q-CDCs is calculated. It is greater than 0.99926 for any prime power q and any δ ≥ 3.

Optimal rank-metric codes in Ferrers diagrams can be used to construct good subspace codes. Such codes consist of matrices having zeros at certain fixed positions. This paper generalizes the known constructions for Ferrers diagram rank-metric (FDRM) codes. Via a criterion for linear maximum rank distance (MRD) codes, an explicit construction for a class of systematic MRD codes is presented, which is used to produce new optimal FDRM codes. By exploring the subcodes of Gabidulin codes, if each of the rightmost$\delta -1$columns in the Ferrers diagram$\cal F$has at least$n-r$dots, where$r$is taken in a range, then the conditions that an FDRM code in$\cal F$is optimal are established. The known combining constructions for FDRM code are generalized by introducing the concept of proper combinations of Ferrers diagrams.

Optical orthogonal codes are commonly used as signature codes for optical code-division multiple access systems. So far, research on 2-D optical orthogonal codes has mainly concentrated on the same autocorrelation and cross-correlation constraints. In this paper, we are concerned about optimal 2-D optical orthogonal codes with the autocorrelation λaand the cross-correlation 1. Some combinatorial constructions for 2-D (n×m,k,λa,1) -optical orthogonal codes are presented. Whenk=3 and λa=2, the exact number of codewords of an optimal 2-D (n×m,3,2,1)-optical orthogonal code is determined for any positive integersn≡ 0,1,3,6,9,10 (mod 12) andm≡ 2(mod 4).

In this paper, we are concerned about optimal two-dimensional optical orthogonal codes with λ = 2 . Some combinatorial constructions are presented and many infinite families of optimal two-dimensional optical orthogonal codes with weight 4 and λ = 2 are obtained. Especially, we shall see that in many cases an optimal two-dimensional optical orthogonal code can not achieve the Johnson bound.