Post on 22-Dec-2015
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
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.
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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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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 .