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 F. The Reed-Muller code
RMF(n,d) is defined by n-variate degree-d
polynomials over F. In this work, we study the list decoding radius
of Reed-Muller codes over a constant prime field F=Fp,
constant degree d and large n. We show that the list decoding radius is
equal to the minimal distance of the code.
That is, if we denote by δ(d) the normalized minimal distance of
RMF(n,d), then the number of codewords in any ball of
radius δ(d)−ε is bounded by c=c(p,d,ε) independent
of n. This resolves a conjecture of Gopalan-Klivans-Zuckerman [STOC 2008],
who among other results proved it in the special case of
F=F2; and extends the work of Gopalan [FOCS 2010] who
proved the conjecture in the case of d=2.
We also analyse the number of codewords in balls of radius exceeding the
minimal distance of the code. For e≤d, we show that the number of
codewords of RMF(n,d) in a ball of radius δ(e)−ε is bounded by exp(c⋅nd−e), where
c=c(p,d,ε) is independent of n. The dependence on n is tight.
This extends the work of Kaufman-Lovett-Porat [IEEE Inf. Theory 2012] who
proved similar bounds over F2.
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