Abstract
Two codewords (a1,...,ak) and (b1,...,bk) form a reverse-free pair if (ai,aj) is not equal to (bj,bi) whenever 1{i,j{k are indices such that ai is not equal to 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.