Incremental codes

We introduce the notion of incremental codes. Unlike a regular code of a given rate, which is an unordered set of elements with a large minimum distance, an incremental code is an ordered vector of elements each of whose prefixes is a good regular code (of the corresponding rate). Additionally, while the quality of a regular code is measured by its minimum distance, we measure the quality of an incremental code C by its competitive ratio A: the minimum distance of each prefix of C has to be at most a factor of A smaller than the minimum distance of the best regular code of the same rate. We first consider incremental codes over an arbitrary compact metric space M, and construct a 2-competitive code for M. When M is finite, the construction takes time O(|M|²), exhausts the entire space, and is NP-hard to improve in general. We then concentrate on 2 specific spaces: the real interval [0, 1] and, most importantly, the Hamming space Fn. For the interval [0, 1] we construct an optimal (infinite) code of competitive ratio ln 4 ≈ 1.386. For the Hamming space Fⁿ (where the generic 2-competitive constructive is not efficient), we show the following. If |F| ≥ q, we construct optimal (and efficient) 1-competitive code that exhausts Fn (has rate 1). For small alphabets (|F| < q), we show that 1-competitive codes do not exist and provide several efficient constructions of codes achieving constant competitive ratios. In particular, our best construction has rate (1−o(1)) and competitive ratio (2+o(1)), essentially matching the bounds in the generic construction.