A mapping of k-bit strings into n -bit strings is called an (α,β)-map if k-bit strings which are more than αk apart are mapped to n-bit strings that are more than βn apart in Hamming distance. This is a relaxation of the classical problem of constructing error-correcting codes, which corresponds to 0579c8548e73fdd0b138e22" title="Click to view the MathML source">α=0. Existence of an (α,β)-map is equivalent to existence of a graph homomorphism , where H(n,d) is a Hamming graph with vertex set 5e23bfcbbbbe454d496e3001" title="Click to view the MathML source">{0,1}n and edges connecting vertices differing in d or fewer entries.
This paper proves impossibility results on achievable parameters (α,β) in the regime of n,k→∞ with a fixed ratio 050">. This is done by developing a general criterion for existence of graph-homomorphism based on the semi-definite relaxation of the independence number of a graph (known as the Schrijver's θ-function). The criterion is then evaluated using some known and some new results from coding theory concerning the θ -function of Hamming graphs. As an example, it is shown that if 05" title="Click to view the MathML source">β>1/2 and – integer, the -fold repetition map achieving 5e784d65b46d6ad4df2ffb" title="Click to view the MathML source">α=β is asymptotically optimal.
050">Finally, constraints on configurations of points and hyperplanes in projective spaces over 0557" title="Click to view the MathML source">F2 are derived.
NGLC 2004-2010.National Geological Library of China All Rights Reserved.
Add:29 Xueyuan Rd,Haidian District,Beijing,PRC. Mail Add: 8324 mailbox 100083
For exchange or info please contact us via email.