### Abstract

Two codewords (a_{1},..., a_{k}) and (b_{1},..., b_{k}) form a reverse-free pair if (a_{i}, a_{j})≠(b_{j}, b_{i}) holds whenever 1≤i<j≤k are indices such that a_{i}≠a_{j}. 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.

Original language | English |
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Pages (from-to) | 383-387 |

Number of pages | 5 |

Journal | Electronic Notes in Discrete Mathematics |

Volume | 38 |

DOIs | |

Publication status | Published - Dec 1 2011 |

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### Keywords

- Codes
- Extremal combinatorics
- Permutations
- Reverse-free

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Electronic Notes in Discrete Mathematics*,

*38*, 383-387. https://doi.org/10.1016/j.endm.2011.09.062