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.

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$r$.

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).