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

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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

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

Presented by

Tian Sang

Jed Liu

2003 March 3rd


Error reduction codes

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

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


Recursive construction of c k

Recursive construction of Ck


Linear time encodable and decodable error correcting codes daniel a spielman

  • 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


Linear time encodable and decodable error correcting codes daniel a spielman

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)


Linear time encodable and decodable error correcting codes daniel a spielman

  • 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

A simple construction

B is a (d, 2d) regular graph


Linear time encodable and decodable error correcting codes daniel a spielman

  • 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

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.


Linear time encodable and decodable error correcting codes daniel a spielman

  • 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


Linear time encodable and decodable error correcting codes daniel a spielman

  • 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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