A cross-bifix-free code of length $n$ over $\mathbb{Z}_q$ is defined as a non-empty subset of $\mathbb{Z}_q^n$ satisfying that the prefix set of each codeword is disjoint from the suffix set of every codeword. Cross-bifix-free codes have found important applications in digital communication systems. One of the main research problems on cross-bifix-free codes is to construct cross-bifix-free codes as large as possible in size. Recently, Wang and Wang introduced a family of cross-bifix-free codes $S_{I,J}^{(k)}(n)$, which is a generalization of the classical cross-bifix-free codes studied early by Lvenshtein, Gilbert and Chee {\it et al.}. It is known that $S_{I,J}^{(k)}(n)$ is nearly optimal in size and $S_{I,J}^{(k)}(n)$ is non-expandable if $k=n-1$ or $1\leq k<n/2$. In this paper, we first show that $S_{I,J}^{(k)}(n)$ is non-expandable if and only if $k=n-1$ or $1\leq k<n/2$, thereby improving the results in [Chee {\it et al.}, IEEE-TIT, 2013] and [Wang and Wang, IEEE-TIT, 2022]. We then construct a new family of cross-bifix-free codes $U^{(t)}_{I,J}(n)$ to expand $S_{I,J}^{(k)}(n)$ such that the resulting larger code $S_{I,J}^{(k)}(n)\bigcup U^{(t)}_{I,J}(n)$ is a non-expandable cross-bifix-free code whenever $S_{I,J}^{(k)}(n)$ is expandable. Finally, we present an explicit formula for the size of $S_{I,J}^{(k)}(n)\bigcup U^{(t)}_{I,J}(n)$.

翻译：定义在 $\mathbb{Z}_q$ 上的长度为 $n$ 的交叉双缀自由码是 $\mathbb{Z}_q^n$ 的一个非空子集，其中每个码字的前缀集与任意码字的后缀集互不相交。交叉双缀自由码在数字通信系统中具有重要应用。关于该类码的核心研究问题之一是如何构造尽可能大的码集。近期，Wang 和 Wang 引入了一族交叉双缀自由码 $S_{I,J}^{(k)}(n)$，该码是早期由 Lvenshtein、Gilbert 及 Chee 等人研究的经典交叉双缀自由码的推广。已知 $S_{I,J}^{(k)}(n)$ 在码字规模上近乎最优，且当 $k=n-1$ 或 $1\leq k<n/2$ 时 $S_{I,J}^{(k)}(n)$ 不可扩展。本文首先证明 $S_{I,J}^{(k)}(n)$ 不可扩展当且仅当 $k=n-1$ 或 $1\leq k<n/2$，从而改进了 [Chee 等， IEEE-TIT， 2013] 及 [Wang 和 Wang， IEEE-TIT， 2022] 中的结论。随后，我们构造了一族新的交叉双缀自由码 $U^{(t)}_{I,J}(n)$ 用于扩展 $S_{I,J}^{(k)}(n)$，使得当 $S_{I,J}^{(k)}(n)$ 可扩展时，其与 $U^{(t)}_{I,J}(n)$ 的并集构成不可扩展交叉双缀自由码。最后，我们给出了 $S_{I,J}^{(k)}(n)\bigcup U^{(t)}_{I,J}(n)$ 码字规模的显式公式。