Linear-time encodable and decodable error-correcting codes Daniel A. Spielman

1. Linear-time encodable and decodable error-correcting codes Daniel A. Spielman Presented by Tian Sang Jed Liu 2003 March 3rd

2. Error-Reduction Codes • Weaker than error-correcting codes • Can remove most of the errors, if not too many message bits and check bits are corrupted • Definition A code C of length n with rn message bits and (1-r)n check bits is an error-reduction code of rate r, error reductionε, and reducible distanceδ, if there is an algorithm, when given a codeword with v ≤ δn corrupt message bits and t ≤δn corrupt check bits, will output a word that differs from the uncorrupted message in at mostεt message bits

3. Error-correcting codes from error-reduction codes • C0: an error-correcting code of block length n0 , rate ¼, andδ/4 fraction of errors can be corrected • Rk: a family of error-reduction codes with n02k message bits, n02k-1 check bits,ε> ½, andδ> 0 • Ck: block lengths n02k and rate ¼ • Mk: the n02k-2 message bits of Ck • Ak: the n02k-3 check bits of encoding Mk using Rk-2 • Bk: the 3n02k-3 check bits of encoding Ak using Ck-1 • C’k: n02k-2 check bits of encoding AkU Bk using Rk-1

4. Recursive construction of Ck

5. Lemma 2 (1) The codes Ck are error-correcting codes of block length n02k and rate ¼ from whichδ/4 fraction of errors can be corrected (2) Ck are linear time encodable/decodable if Rk have linear time encoding algorithm and linear time error-reduction algorithm that will (a) on input a word with corrupt messagebitsand check bits v, t ≤δn, output a word with at most max(v/2, t/2) corrupt message bits (b) If v ≤δn and t = 0, output the codeword without corrupt bits

6. Proof by induction Base case is the code C0 of block length n0 , rate ¼, andδ/4 fraction of errors can be corrected. Obviously we can encode/decode C0 in constant time c • Encoding time of Ck According to the assumption, Rk is linear time encodable/reductable, say c1n02k, c2n02k respectively The time to encode Ck = the time to encode Rk-2 + the time to encode Ck-1 +the time to encode Rk-1 =c1n02k-2 + (3c1n02k-2 + c) + c1n02k-1 =3c1n02k-1 + c (linear in the size of Ck)

7. Error-correction of Ck There are not many errors in Ck(≤δ/4 fraction) (1) Use C’k as check bits to reduce errors in AkUBk, then not many errors left in AkUBk(≤δ/8 fraction) (2) In fact is a AkUBk codeword of Ck-1, by inductive hypothesis, Ck-1 can correct all left errors in AkUBk (3) Since now Akis free of error, and not many errors in Mk(≤δ/4 fraction), we can use Ak as check bits to correct all errors in Mk (according to the assumption at (2)(b))

8. A simple construction B is a (d, 2d) regular graph

9. Simple Sequential Error-Reduction Algorithm Repeat If there is a message bit that has more unsatisfied than satisfied neighbors, then flip that bit Until no such message bit remains • Lemma 10 let B be a (c,d,α,3/4 d + 2) expander graph, if the algorithm above for R(B) is given a word x that differs from a codeword w of R(B) in at most v≤αn/2message bits and t ≤αn/2check bits, then the algorithm will output a word that differs from w in at most t/2 of its message bits

10. Proof • This algorithm is very similar to the simple sequential algorithm for expander codes • First show ifαn≥ v≥ t/2, there is a node that has more unsatisfied than satisfied neighbors • Since each time the number of unsatisfied check bits decreases, we can prove αn ≥ v is always true. So the algorithm can only end up with v <t/2, which means errors are reduced. • Constant degrees, obviously in linear time.

11. Simple Parallel Error-Reduction Round • For each message bit, count the number of unsatisfied check bits among its neighbors • Flip each message bit that has more unsatisfied than satisfied neighbors • Lemma 13 Assume a word differs from a codeword w of R(B) in at most v≤αn/2message bits and t ≤αn/2check bits, then the round algorithm will output a word that differs from w in at most v(d-4)/d of its message bits • Simple Parallel Error-Reduction Algorithm Iterate logd/(d-4)2 simple parallel error-reduction rounds

12. Theorem 15 From a family of (c,d,α,3/4 d + 4) expander graphs between sets of n02k and n02k-1 vertices for all k ≥-1, one can construct an infinite family of error-correcting Codes that have linear-time encoding algorithms and linear-time decoding algorithms that will correct an α/8 Fraction of error. • Problem Such graphs can be only obtained through a randomized construction.