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Almost Sure Convergence of Relative Frequency of Occurrence of Burst Errors on Channels with Memory
Mitsuru HAMADA
Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences
Vol.E82A
No.10
pp.20222033 Publication Date: 1999/10/25
Online ISSN:
DOI:
Print ISSN: 09168508 Type of Manuscript: Special Section PAPER (Special Section on Information Theory and Its Applications) Category: Coding Theory Keyword: bursterrorcorrecting code, burst metric (weight), Markov chain, law of large numbers, swept covering,
Full Text: PDF(452KB)>>
Summary:
Motivated by intention to evaluate asymptotically multiplebursterrorcorrecting codes on channels with memory, we will derive the following fact. Let {Z_{i} } be a hidden Markov process, i. e. , a functional of a Markov chain with a finite state space, and W_{b}(Z_{1}Z_{2}Z_{n}) denote the number of burst errors that appear in Z_{1}Z_{2}Z_{n}, where the number of burst errors is counted using Gabidulin's burst metric , 1971. As the main result, we will prove the almost sure convergence of relative burst weight W_{b}(Z_{1}Z_{2}Z_{n})/n, i. e. , the relative frequency of occurrence of burst errors, for a broad class of functionals { Z_{i} } of finite Markov chains. Functionals of Markov chains are often adopted as models of the noises on channels, especially on burstnoise channels, the most famous model of which is probably the Gilbert channel proposed in 1960. Those channel models described with Markov chains are called channels with memory (including channels with zeromemory, i. e. , memoryless ones). This work's achievement enables us to extend Gilbert's code performance evaluation in 1952, a landmark that offered the wellknown Gilbert bound, discussed its relationship to the (memoryless) binary symmetric channel, and has been serving as a guide for theHammingmetricbased design of errorcorrecting codes, to the case of theburstmetricbased codes (bursterrorcorrecting codes) and discrete channels with or without memory.

