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Erschienen in:
2009 | OriginalPaper | Buchkapitel
Succinct Representation of Codes with Applications to Testing
verfasst von : Elena Grigorescu, Tali Kaufman, Madhu Sudan
Erschienen in: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Verlag: Springer Berlin Heidelberg
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Motivated by questions in property testing, we search for linear error-correcting codes that have the “single local orbit” property: they are specified by a single local constraint and its translations under the symmetry group of the code. We show that the dual of every “sparse” binary code whose coordinates are indexed by elements of
${\mathbb{F}}_{2^n}$
for prime
n
, and whose symmetry group includes the group of non-singular affine transformations of
${\mathbb{F}}_{2^n}$
, has the single local orbit property. (A code is
sparse
if it contains polynomially many codewords in its block length.) In particular this class includes the dual-BCH codes for whose duals (BCH codes) simple bases were not known. Our result gives the first short (
O
(
n
)-bit, as opposed to
$\exp(n)$
-bit) description of a low-weight basis for BCH codes. If 2
n
− 1 is a Mersenne prime, then we get that every sparse
cyclic
code also has the single local orbit.