NDLI: Linearity testing in characteristic two

Content Provider	IEEE Xplore Digital Library
Author	Bellare, M. Coppersmith, D. Hastad, J. Kiwi, M. Sudan, M.
Copyright Year	1963
Abstract	Let Dist(f,g)=Pr/sub u/[f(u)/spl ne/g(u)] denote the relative distance between functions f,g mapping from a group G to a group H, and let Dist(f) denote the minimum, over all linear functions (homomorphisms) g, of Dist(f,g). Given a function f:G/spl rarr/H we let Err(f)=Pr/sub u,/spl upsi//[f(u)+f(/spl upsi/)/spl ne/f(u+/spl upsi/)] denote the rejection probability of the Blum-Luby-Rubinfeld (1993) linearity test. Linearity testing is the study of the relationship between Err(f) and Dist(f), and in particular lower bounds on Err(f) in terms of Dist(f). We discuss when the underlying groups are G=GF(2)/sup n/ and H=GF(2). In this case, the collection of linear functions describe a Hadamard code of block length 2/sup n/ and for an arbitrary function f mapping GF(2)/sup n/ to GF(2) the distance Dist(l) measures its distance to a Hadamard code. Err(f) is a parameter that is "easy to measure" and linearity testing studies the relationship of this parameter to the distance of f. The code and corresponding test are used in the construction of efficient probabilistically checkable proofs and thence in the derivation of hardness of approximation. Improved analyses translate into better nonapproximability results. We present a description of the relationship between Err(f) and Dist(f) which is nearly complete in all its aspects, and entirely complete in some. We present functions L,U:[0,1]/spl rarr/[0,1] such that for all x /spl isin/ [0,1] we have L(x)/spl les/Err(f)/spl les/U(x) whenever Dist(f)=x, with the upper bound being tight on the whole range, and the lower bound tight on a large part of the range and close on the rest. Part of our strengthening is obtained by showing a new connection between linearity testing and Fourier analysis.
Sponsorship	IEEE Information Theory Society
Starting Page	1781
Ending Page	1795
Page Count	15
File Size	1671923
File Format	PDF
ISSN	00189448
Volume Number	42
Issue Number	6
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	1996-11-01
Publisher Place	U.S.A.
Access Restriction	One Nation One Subscription (ONOS)
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Linearity Testing Computer science Error correction Error correction codes Length measurement Upper bound Computer errors Scholarships Mathematics
Content Type	Text
Resource Type	Article
Subject	Library and Information Sciences Information Systems Computer Science Applications

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