Explicit low-weight bases for BCH codes

Elena Grigorescu, Tali Kaufman

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

Abstract

We exhibit explicit bases for BCH codes of designed distance 5. While BCH codes are some of the most studied families of codes, only recently Kaufman and Litsyn (FOCS, 2005) showed that they admit bases of small weight codewords. Furthermore, Grigorescu, Kaufman, and Sudan (RANDOM, 2009) and Kaufman and Lovett (FOCS, 2011) proved that, in fact, BCH codes can admit very structured bases of small weight codewords (i.e., bases that can be fully specified by a single codeword and its orbit under the affine group). The existence of such structured bases has applications in property testing, and motivates our search for a fully explicit description of low weight codewords and, in particular, of codewords that generate a basis for BCH codes. In this paper, we describe the support of basis-generating codewords under affine transformations of the domain for the very specific case of binary (extended) ${\rm BCH}(2, n)$. We believe that extending these findings to general BCH codes merits further investigation.

Original languageEnglish
Article number6017120
Pages (from-to)78-81
Number of pages4
JournalIEEE Transactions on Information Theory
Volume58
Issue number1
DOIs
StatePublished - Jan 2012

Bibliographical note

Funding Information:
Manuscript received December 17, 2010; revised August 31, 2011; accepted September 02, 2011. Date of publication September 12, 2011; date of current version January 06, 2012. This work appeared in E. Grigorescu’s Ph.D. dissertation. Research for this work was conducted when the authors were with MIT CSAIL. E. Grigorescu was supported in part by NSF Grant CCR-0829672 and in part by NSF Award 1019343 to the Computing Research Association for the CI Fellows Project. T. Kaufman was supported in part by NSF Grant CCR-0829672 and in part by the Alon Fellowship.

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