Succinct representation of codes with applications to testing

Elena Grigorescu; Tali Kaufman; Madhu Sudan

doi:10.1007/978-3-642-03685-9_40

Succinct representation of codes with applications to testing

Elena Grigorescu, Tali Kaufman, Madhu Sudan

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

16 Scopus citations

Abstract

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 double-struck F _2n for prime n, and whose symmetry group includes the group of non-singular affine transformations of double-struck F_2n, 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 -bit) description of a low-weight basis for BCH codes. If 2ⁿ - 1 is a Mersenne prime, then we get that every sparse cyclic code also has the single local orbit.

Original language	English
Title of host publication	Approximation, Randomization, and Combinatorial Optimization
Subtitle of host publication	Algorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings
Pages	534-547
Number of pages	14
DOIs	https://doi.org/10.1007/978-3-642-03685-9_40
State	Published - 2009
Externally published	Yes
Event	12th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2009 and 13th International Workshop on Randomization and Computation, RANDOM 2009 - Berkeley, CA, United States Duration: 21 Aug 2009 → 23 Aug 2009

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	5687 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	12th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2009 and 13th International Workshop on Randomization and Computation, RANDOM 2009
Country/Territory	United States
City	Berkeley, CA
Period	21/08/09 → 23/08/09

Keywords

Affine/cyclic invariance
Locally testable codes
Single orbit

Access to Document

10.1007/978-3-642-03685-9_40

Cite this

Grigorescu, E., Kaufman, T., & Sudan, M. (2009). Succinct representation of codes with applications to testing. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings (pp. 534-547). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5687 LNCS). https://doi.org/10.1007/978-3-642-03685-9_40

Grigorescu, Elena ; Kaufman, Tali ; Sudan, Madhu. / Succinct representation of codes with applications to testing. Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings. 2009. pp. 534-547 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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abstract = "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 double-struck F 2n for prime n, and whose symmetry group includes the group of non-singular affine transformations of double-struck F2n, 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 -bit) description of a low-weight basis for BCH codes. If 2n - 1 is a Mersenne prime, then we get that every sparse cyclic code also has the single local orbit.",

keywords = "Affine/cyclic invariance, Locally testable codes, Single orbit",

author = "Elena Grigorescu and Tali Kaufman and Madhu Sudan",

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Grigorescu, E, Kaufman, T & Sudan, M 2009, Succinct representation of codes with applications to testing. in Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 5687 LNCS, pp. 534-547, 12th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2009 and 13th International Workshop on Randomization and Computation, RANDOM 2009, Berkeley, CA, United States, 21/08/09. https://doi.org/10.1007/978-3-642-03685-9_40

Succinct representation of codes with applications to testing. / Grigorescu, Elena; Kaufman, Tali; Sudan, Madhu.
Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings. 2009. p. 534-547 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5687 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Grigorescu E, Kaufman T, Sudan M. Succinct representation of codes with applications to testing. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings. 2009. p. 534-547. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-642-03685-9_40

Succinct representation of codes with applications to testing

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