All binary linear codes that are invariant under PSL2(n)

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The projective special linear group PSL 2 (n) is 2-transitive for all primes n and 3-homogeneous for n = 3 (mod 4) on the set (0, 1, ... , n - 1, ∞). It is known that the extended odd-like quadratic residue codes are invariant under PSL 2 (n). Hence, the extended quadratic residue codes hold an infinite family of 2-designs for primes n = 1 (mod 4), an infinite family of 3-designs for primes n = 3 (mod 4). To construct more t-designs with t ∈ (2, 31), one would search for other extended cyclic codes over finite fields that are invariant under the action of 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 PSL 2 (n).

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© 2017 IEEE. Publisher’s version of record: https://dx.doi.org/10.1109/TIT.2017.2746860

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IEEE Transactions on Information Theory