On metric properties of maps between Hamming spaces and related graph homomorphisms
Name
1503.02779.pdf
Description
Submitted version
Size
330.27 KB
Format
Adobe PDF
Checksum (MD5)
2d784b70db9a76f6344f630de3a3b79b
Author(s)
Polyanskiy, Yury
Date Issued
January 2017
Journal
Journal of Combinatorial Theory, Series A
Publisher
Elsevier BV
Citation
Polyanskiy, Yury. "On metric properties of maps between Hamming spacesand related graph homomorphisms." Journal of Combinatorial Theory, Series A, 145 (Jan. 2017): pages 227-251.
Version
Original manuscript
Abstract
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 α=0. Existence of an (α,β)-map is equivalent to existence of a graph homomorphism H¯(k,αk)→H¯(n,βn), where H(n,d) is a Hamming graph with vertex set {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 nk=ρ. 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 β>1/2 and nk – integer, the nk-fold repetition map achieving α=β is asymptotically optimal. Finally, constraints on configurations of points and hyperplanes in projective spaces over F2 are derived.
Subjects
Theoretical Computer Science
Computational Theory and Mathematics
Discrete Mathematics and Combinatorics
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Terms of Use
Creative Commons Attribution-NonCommercial-NoDerivs License
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1016/j.jcta.2016.08.005
