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

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

Presented by

Tian Sang

Jed Liu

2003 March 3rd

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

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

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

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)

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))

A simple construction

B is a (d, 2d) regular graph

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

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
- Constant degrees, obviously in linear time.

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

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.

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