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 language | English |
|---|---|
| Article number | 6017120 |
| Pages (from-to) | 78-81 |
| Number of pages | 4 |
| Journal | IEEE Transactions on Information Theory |
| Volume | 58 |
| Issue number | 1 |
| DOIs | |
| State | Published - 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.
Funding
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.
| Funders | Funder number |
|---|---|
| National Science Foundation | 1019343, CCR-0829672 |