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Coding Theory : A General Framework and Two Inverse Problems

Bortolussi L., Dinu L. P., Franzoi L., Sgarro A.

Fundamenta Informaticae

2015

Vol. 141, nr 4

297--310

We put forward an ample framework for coding based on upper probabilities, or more generally on normalized monotone set-measures, and model accordingly noisy transmission channels and decoding errors. Two inverse problems are considered. In the first case, a decoder is given and one looks for channels of a specified family over which that decoder would work properly. In the second and more ambitious case, it is codes which are given, and one looks for channels over which those codes would ensure the required error correction capabilities. Upper probabilities allow for a solution of the two inverse problems in the case of usual codes based on checking Hamming distances between codewords: one can equivalently check suitable upper probabilities of the decoding errors. This soon extends to “odd” codeword distances for DNA strings as used in DNA word design, where instead, as we prove, not even the first unassuming inverse problem admits of a solution if one insists on channel models based on ”usual” probabilities.

Spearman Permutation Distances and Shannon's Distinguishability

Bortolussi L., Dinu L.P., Sgarro A.

Fundamenta Informaticae

2012

Vol. 118, nr 3

245-252

Spearman distance is a permutation distance which might be used for codes in permutations beside Kendall distance. However, Spearman distance gives rise to a geometry of strings, which is rather unruly from the point of view of error correction and error detection. Special care has to be taken to discriminate between the two notions of codeword distance and codeword distinguishability. This stresses the importance of rejuvenating the latter notion, extending it from Shannon's zero-error information theory to the more general setting of metric string distances.

A Low-complexity Distance for DNA Strings

Dinu L.P., Sgarro A.

Fundamenta Informaticae

2006

Vol. 73, nr 3

361-372

We exhibit a low-complexity but non-trivial distance between strings to be used in biology. The experimental results we provide were obtained on a standard laptop and, even if preliminary, are quite encouraging.