Linear-Time Encodable and Decodable Error-Correcting Codes

Jed Liu3 March 2003

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

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

The construction

.

.

.

Ci = S(mi1 , mi2

, …, mid)

m2

m3

m4

mn

m1

B

R(B,S)(m) = C1, C2, …, C(c/d)n

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.

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

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 .

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.

2329

51

22 dnd

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.

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

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.

Instantiating the variables

Let G = {Gni,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.

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51

2 d

d

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 .

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.