Introduction to coding theory (CMU, Spr 2010)

April 30, 2010

Lecture 26 summary

Filed under: Lecture summary — Venkat Guruswami @ 5:17 pm

We mentioned two topics which were introduced to coding theory by theoretical computer science: local testing and local decoding of codes. These and related topics (such as PCPs and applications of locally decodable codes in complexity and cryptography) have been intensively researched in the last 10-15 years, with several breakthroughs occurring in recent years.

We focused on local (unique) decoding of codes for the lecture. We saw how Hadamard codes can be locally decoded using just two queries. However, their encoding length for a message of length n is 2^n. We then saw the higher degree generalization of Hadamard codes, where the message is interpreted as a degree D homogeneous multilinear polynomial (i.e., all terms have degree exactly D \ge 2). This gave us codes of encoding length \approx 2^{O(n^{1/D})}, and we discussed a 2D-query local decoding algorithm. This was based on interpolating the restriction of the multilinear polynomial on a line in a random direction. Thus for any constant q, we got codes that are locally decodable using q queries that have encoding length 2^{n^O(1/q)}.

We then turned to the ingenious 3-query locally decodable code (LDC) construction due to Yekhanin. In keeping with the theme of our initial constructions, we presented a polynomial view of these codes, where the messages are again interpreted as homegeneous multilinear polynomials of certain degree (say D) but only a carefully chosen subset of all possible {M \choose D} monomials are allowed. (This actually reduces the rate compared to our earlier construction, but the big gain is that one is able to locally decode using only three queries instead of about D queries!) Our description is based on a variant of Yekhanin’s construction that was discovered by Raghavendra and subsequently presented by Gopalan as polynomial based codes.

For every t such that 2^t-1 is prime (such a prime is called a Mersenne prime), we gave a construction of 3-query LDCs  of encoding length \exp(O_t(n^{1/t})). Since very large Mersenne primes are known, we get 3-query LDCs of encoding length less than \exp(O(n^{10^{-7}})). We presented a 3-query algorithm and proved its correctness assuming the stated properties of the “matching sets” U_i,V_i used in the construction, and then explained how to construct families of such subsets of \{1,2,\dots,M\} of size \Omega_t(M^t).

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