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Infinite Families of Linear Codes Supporting More t -Designs.
- Source :
-
IEEE Transactions on Information Theory . Jul2022, Vol. 68 Issue 7, p4365-4377. 13p. - Publication Year :
- 2022
-
Abstract
- Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of BCH codes $\mathcal {C}_{(q,q+1,4,1)}$ and their dual codes with $q=2^{m}$ and established that the codewords of the minimum (or the second minimum) weight in these codes support 4-designs or 3-designs. Motivated by this, we further investigate the codewords of the next adjacent weight in such codes and discover more infinite classes of $t$ -designs with $t=3,4$. In particular, we prove that codewords of weight 7 in $\mathcal {C}_{(q,q+1,4,1)}$ support 4-designs for odd $m \geqslant 5$ and they support 3-designs for even $m \geqslant 4$ , which provide infinite classes of simple $t$ -designs with new parameters. Another significant class of $t$ -designs we produce in this paper has complementary designs with parameters 4- $(2^{2s+1}+ 1,5,5)$ ; these designs have the smallest index among all the known simple 4- $(q+1,5,\lambda)$ designs derived from codes for prime powers $q$ ; and they are further proved to be isomorphic to the 4-designs admitting the projective general linear group PGL $(2,2^{2s+1})$ as the automorphism group constructed by Alltop in 1969. [ABSTRACT FROM AUTHOR]
- Subjects :
- *AUTOMORPHISM groups
*CYBERNETICS
*AUTOMORPHISMS
*LINEAR codes
Subjects
Details
- Language :
- English
- ISSN :
- 00189448
- Volume :
- 68
- Issue :
- 7
- Database :
- Academic Search Index
- Journal :
- IEEE Transactions on Information Theory
- Publication Type :
- Academic Journal
- Accession number :
- 157551873
- Full Text :
- https://doi.org/10.1109/TIT.2022.3156072