2017/04/04 by Cunsheng Ding, Hao Liu, Ding, Cunsheng +3
Computer Science · Mathematics · #FOS: Computer and information sciences #Information Theory (cs.IT) #cs.IT #math.IT
paper · pdf · doi:10.48550/arxiv.1704.01199
arxiv created 2017/04/04 · arxiv updated 2017/04/06
The projective special linear group \PSL2(n) is 2-transitive for all primes n and 3-homogeneous for n ≡ 3 \pmod4 on the set \0,1, ⋯, n-1, ∞\. It is known that the extended odd-like quadratic residue codes are invariant under \PSL2(n). Hence, the extended quadratic residue codes hold an infinite family of 2-designs for primes n ≡ 1 \pmod4, an infinite family of 3-designs for primes n ≡ 3 \pmod4. To construct more t-designs with t ∈ \2, 3\, one would search for other extended cyclic codes over finite fields that are invariant under the action of \PSL2(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 \PSL2(n).