On non-expandable cross-bifix-free codes
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A cross-bifix-free code of length n over ℤ_q is defined as a non-empty subset of 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 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≤ 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≤ k<n/2, thereby improving the results in [Chee 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)⋃ 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)⋃ U^(t)_I,J(n).
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