337 research outputs found
List decoding Reed-Muller codes over small fields
The list decoding problem for a code asks for the maximal radius up to which
any ball of that radius contains only a constant number of codewords. The list
decoding radius is not well understood even for well studied codes, like
Reed-Solomon or Reed-Muller codes.
Fix a finite field . The Reed-Muller code
is defined by -variate degree-
polynomials over . In this work, we study the list decoding radius
of Reed-Muller codes over a constant prime field ,
constant degree and large . We show that the list decoding radius is
equal to the minimal distance of the code.
That is, if we denote by the normalized minimal distance of
, then the number of codewords in any ball of
radius is bounded by independent
of . This resolves a conjecture of Gopalan-Klivans-Zuckerman [STOC 2008],
who among other results proved it in the special case of
; and extends the work of Gopalan [FOCS 2010] who
proved the conjecture in the case of .
We also analyse the number of codewords in balls of radius exceeding the
minimal distance of the code. For , we show that the number of
codewords of in a ball of radius is bounded by , where
is independent of . The dependence on is tight.
This extends the work of Kaufman-Lovett-Porat [IEEE Inf. Theory 2012] who
proved similar bounds over .
The proof relies on several new ingredients: an extension of the
Frieze-Kannan weak regularity to general function spaces, higher-order Fourier
analysis, and an extension of the Schwartz-Zippel lemma to compositions of
polynomials.Comment: fixed a bug in the proof of claim 5.6 (now lemma 5.5
Short seed extractors against quantum storage
Some, but not all, extractors resist adversaries with limited quantum
storage. In this paper we show that Trevisan's extractor has this property,
thereby showing an extractor against quantum storage with logarithmic seed
length
Optimal Iris Fuzzy Sketches
Fuzzy sketches, introduced as a link between biometry and cryptography, are a
way of handling biometric data matching as an error correction issue. We focus
here on iris biometrics and look for the best error-correcting code in that
respect. We show that two-dimensional iterative min-sum decoding leads to
results near the theoretical limits. In particular, we experiment our
techniques on the Iris Challenge Evaluation (ICE) database and validate our
findings.Comment: 9 pages. Submitted to the IEEE Conference on Biometrics: Theory,
Applications and Systems, 2007 Washington D
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