Two codewords (a1,..., ak) and (b1,..., bk) form a reverse-free pair if (ai, aj)≠(bj, bi) holds whenever 1≤i<j≤k are indices such that ai≠aj. In a reverse-free code, each pair of codewords is reverse-free. The maximum size of a reverse-free code with codewords of length k and an n-element alphabet is denoted by F (n,k). Let F(n, k) denote the maximum size of a reverse-free code with all codewords consisting of distinct entries.We determine F (n,3) and F (n,3) exactly whenever n is a power of 3, and asymptotically for other values of n. We prove non-trivial bounds for F(n, k) and F (n,k) for general k and for other related functions as well. Using VC-dimension of a matrix, we determine the order of magnitude of F (n,k) for n fixed and k tending to infinity.
- Extremal combinatorics
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Applied Mathematics