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...... C i = S(m i 1, m i 2, …, m i d ) m2m2 m3m3 m4m4 mnmn m1m1 B R (B,S)(m) = C 1, C 2, …, C (c/d)n

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  /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.  is the minimum relative distance of S

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

8. Per-round error reduction Lemma: If an error-reduction round is given an input that differs from a codeword w in at most  dn/2 message bits and at most  dn/2 check bits, then at the end of the round, the word will differ from w in at most message 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  < k  fraction of error, where k  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  is chosen, then by the Gilbert-Varshamov bound, for all large enough block lengths d, there exists a code of minimum relative distance  and rate r = 1 – H(  ) > 4/5. Fix S to be one such code.

12. Instantiating the variables Let G = {G n i,d} be a polytime- constructible dense family of good expander graphs. Let d 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  and , 1/5 + 9(2  +  )/  2 < 1/4.

13. Finishing the construction Let B n i,d be the edge-vertex incidence graph of G n i,d. The family of error- reduction codes consists of the codes R (B n i,d,S). Use the Gilbert-Varshamov bound to find a C 0 of block length n 0, rate ¼, minimum relative distance .

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.