Constructions of Non-Expandable Cross-Bifix-Free Codes via Expandable Codes

A cross-bifix-free code of length $n$ over $\mathbb {Z}_{q}$ is a non-empty subset of $\mathbb {Z}_{q}^{n}$ such that the prefix set of each codeword is disjoint from the suffix set of every codeword. To achieve good performance in communication systems, it is desirable to construct cross-bifix-free codes with large size. Recently, Wang and Wang generalized the classical cross-bifix-free codes presented by Levenshtein, Gilbert and Chee et al. by constructing a new family of cross-bifix-free codes $S_{I,J}^{(k)}(n)$ . The code $S_{I,J}^{(k)}(n)$ is nearly optimal in terms of its size and non-expandable if $k=n-1$ or $1\leq k < n/2$ . There are three major ingredients in this paper. The first is to improve the results in Chee et al. (2013) and Wang and Wang (2022) in which we prove that the code $S_{I,J}^{(k)}(n)$ is non-expandable if and only if $k=n-1$ or $1\leq k < n/2$ . The second ingredient contributes to a new family of cross-bifix-free codes $U^{(t)}_{I,J}(n)$ . This new code enables us to construct non-expandable cross-bifix-free codes $S_{I,J}^{(k)}(n)\bigcup U^{(t)}_{I,J}(n)$ whenever $S_{I,J}^{(k)}(n)$ is expandable. The union of $U^{(t)}_{I,J}(n)$ and $S_{I,J}^{(k)}(n)$ enlarges the size of $S_{I,J}^{(k)}(n)$ . Finally, we give an explicit formula for the size of $S_{I,J}^{(k)}(n)\bigcup U^{(t)}_{I,J}(n)$ .