Some combinatorial constructions for optimal perfect deletion-correcting codes

There are two kinds of perfect t-deletion-correcting codes of length k over an alphabet of size v, those where the coordinates may be equal and those where all coordinates must be different. We call these two kinds of codes T*(k − t, k, v)-codes and T(k − t, k, v)-codes respectively. The cardinality of a T(k − t, k, v)-code is determined by its parameters, while T*(k − t, k, v)-codes do not necessarily have a fixed size. Let N(k − t, k, v) denote the maximum number of codewords in any T*(k − t, k, v)-code. A T*(k − t, k, v)-code with N(k − t, k, v) codewords is said to be optimal. In this paper, some combinatorial constructions for optimal T*(2, k, v)-codes are developed. Using these constructions, we are able to determine the values of N(2, 4, v) for all positive integers v. The values of N(2, 5, v) are also determined for almost all positive integers v, except for v = 13, 15, 19, 27 and 34.