Optimal Cycle Codes Constructed From Ramanujan Graphs
نویسندگان
چکیده
We aim here at showing how some known Ramanujan Cayley graphs yield error-correcting codes that are asymptotically optimal in the class of cycle codes of graphs. The main reason why known constructions of Ramanujan graphs yield good cycle codes is that the number of their cycles of a given length behaves essentially like that of random regular graphs. More precisely we show that for actual constructions of Ramanujan graphs of degree ∆ which are bipartite, and for the double cover of known Ramanujan graphs which are not bipartite, the number of cycles of length 2l is Oε(∆ − 1 + ε)2l (for every ε > 0), which is as about the same as one could expect from a random regular graph of degree ∆. Furthermore, it is possible to show that this property guarantees the highest possible error-probability p that the corresponding cycle codes can sustain, among the class of cycle codes of ∆-regular graphs. This gives a constructive answer to an early problem in coding theory, namely determining what is asymptotically the best possible performance of cycle codes of graphs, when submitted to the binary symmetric channel.
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عنوان ژورنال:
- SIAM J. Discrete Math.
دوره 10 شماره
صفحات -
تاریخ انتشار 1997