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All Binary Linear Codes That Are Invariant Under ${\mathrm{PSL}}_2(n)$.
- Source :
-
IEEE Transactions on Information Theory . Aug2018, Vol. 64 Issue 8, p5769-5775. 7p. - Publication Year :
- 2018
-
Abstract
- The projective special linear group ${\mathrm {PSL}}_{2}(n)$ is 2-transitive for all primes $n$ and 3-homogeneous for $n \equiv 3 \pmod {4}$ on the set $\{0,1, \ldots, n-1, \infty \}$. It is known that the extended odd-like quadratic residue codes are invariant under ${\mathrm {PSL}}_{2}(n)$. Hence, the extended quadratic residue codes hold an infinite family of 2-designs for primes $n \equiv 1 \pmod {4}$ , an infinite family of 3-designs for primes $n \equiv 3 \pmod {4}$. To construct more $t$ -designs with $t \in \{2, 3\}$ , one would search for other extended cyclic codes over finite fields that are invariant under the action of ${\mathrm {PSL}}_{2}(n)$. The objective of this paper is to prove that the extended quadratic residue binary codes are the only nontrivial extended binary cyclic codes that are invariant under ${\mathrm {PSL}}_{2}(n)$. [ABSTRACT FROM AUTHOR]
- Subjects :
- *LINEAR codes
*CYCLIC codes
*BLOCK codes
*RESIDUE codes
*INFORMATION theory
Subjects
Details
- Language :
- English
- ISSN :
- 00189448
- Volume :
- 64
- Issue :
- 8
- Database :
- Academic Search Index
- Journal :
- IEEE Transactions on Information Theory
- Publication Type :
- Academic Journal
- Accession number :
- 130667062
- Full Text :
- https://doi.org/10.1109/TIT.2017.2746860