Linear-Time Encodable and Decodable Error-Correcting Codes

1. Linear-Time Encodable and Decodable Error-Correcting Codes Jed Liu 3 March 2003

2. Explicit constructions • Only randomized constructions known for families of very good expander graphs. • To produce explicit constructions, use explicit constructions of expander graphs. • Idea: use edge-vertex incidence graphs of very good expanders

3. The construction • B is a (c,d)-regular bipartite graph with n left-vertices and (c/d)nk right-vertices. • S is a linear code with d message bits, k check bits. • R(B,S) is an error-reduction code with n message bits and (c/d)nk check bits

4. The construction m1 m2 m3 m4 . . . Ci = S(mi1, mi2, …, mid) R(B,S)(m) = C1, C2, …, C(c/d)n mn B

5. Properties of R(B,S) • Can be encoded in linear time. • Theorem: If B is the edge-vertex incidence graph of a good expander, then R(B,S) is a good error-reduction code.

6. Parallel Error-Reduction Round for R(B,S) • In parallel for each cluster, if check bits in the cluster and the associated message are within e/6 of a codeword: • Send a flip signal to every message bit that differs from the corresponding bit in the codeword. • Any message bit that receives at least one flip signal gets flipped. • e is the minimum relative distance of S

7. Per-round error reduction • S = linear code of rate r, block length d, minimum relative distance e. • B = edge-vertex incidence graph of a d-regular graph on n vertices with second-largest eïgenvalue l.

8. Per-round error reduction • Lemma: If an error-reduction round is given an input that differs from a codeword w in at most adn/2 message bits and at most bdn/2 check bits, then at the end of the round, the word will differ from w in at mostmessage bits.

9. The main theorem • Theorem: There exists a polytime-constructible family of error-correcting codes with rate ¼ and have linear-time encoding and decoding algorithms that can correct any g < ke fraction of error, where ke is a (very) small constant. • The proof makes heavy use of the Gilbert-Varshamov bound.

10. Proving the main theorem • Build the error-correcting codes by constructing a family of error-reduction codes. • The error-reduction codes will be of the form R(B,S). • S = a particular good code known to exist by the Gilbert-Varshamov bound • B = edge-vertex incidence graphs of a dense family of good expander graphs

11. Instantiating the variables • If an appropriate e is chosen, then by the Gilbert-Varshamov bound, for all large enough block lengths d, there exists a code of minimum relative distance e and rate r = 1 – H(e) > 4/5. Fix S to be one such code.

12. Instantiating the variables • Let G = {Gni,d} be a polytime-constructible dense family of good expander graphs. Let ld be the upper bound on the second-largest eïgenvalues of its graphs of degree d. • Fix d so that . • Such a d exists because for small enough a and b, 1/5 + 9(2a+b)/e2 < 1/4.

13. Finishing the construction • Let Bni,d be the edge-vertex incidence graph of Gni,d. The family of error-reduction codes consists of the codes R(Bni,d,S). • Use the Gilbert-Varshamov bound to find a C0 of block length n0, rate ¼, minimum relative distance e.

14. Remarks on the construction • Used Gilbert-Varshimov bound to find S. • Spielman: “A constant amount of nonconstructivity is negligible.” • Instead, can pick S to be any known asymptotically good code, or fix d and pick an appropriate error-correcting code.