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

Authors

Cunsheng Ding, Hong Kong University of Science and Technology
Hao Liu, Hong Kong University of Science and Technology
Vladimir Tonchev, Michigan Technological UniversityFollow

Document Type

Article

Publication Date

8-2018

Abstract

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).

Publisher's Statement

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

Publication Title

IEEE Transactions on Information Theory

Recommended Citation

Ding, C., Liu, H., & Tonchev, V. (2018). All binary linear codes that are invariant under PSL2(n). IEEE Transactions on Information Theory, 64(8), 5769-5775. http://doi.org/10.1109/TIT.2017.2746860
Retrieved from: https://digitalcommons.mtu.edu/math-fp/71