On the symmetry of good nonlinear codes

It is shown that there are arbitrarily long “good” (in the sense of Gilbert) binary block codes that are preserved under very large permutation groups. This result contrasts sharply with the properties of linear codes: it is conjectured that long cyclic codes are bad, and known that long al&e-invariant codes are bad. 1 T IS the object of this paper to show that there exist arbitrarily long binary codes that have large permutation groups and also satisfy the Gilbert bound. Kasami’s result’ that long linear codes of length n = 2”, which admit the affine group of order n (n l), cannot satisfy the Gilbert bound, makes it extremely unlikely that our result can be extended to linear codes. What our result does show, however, is that it cannot be merely the presence of a moderate-sized permutation group that forces a code to be bad, just as linearity does not necessarily degrade performance. The precise statement of our result is given in Theorem 1. Theorem 1 For each n 2 1, suppose P, is a group of permutations on {1,2, ... , n), of order pn, and that each nonidentity permutation in P, has 5 fn fixed points. We assume that logp, = fn = 44, o(n). Then for any 0 < D < l/2, there exists a sequence of codes C,, (C,, has block length n) with rates R,, so that Cn admits P,, as a permutation group; &in (CJ 2 Dn; lim inf R, 2 1 H, (D), where H, (+) is the binary entropy function. For example, Theorem 1 permits the following choices of groups P,. P, = cyclic group of order n. Manuscript received July 23, 1969; revised November 3, 1969. This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, sponsored by the National Aeronautics and Space Administratlon under Contract NAS 7-100. The author is with Jet Propulsion Laboratory, Pasadena, Calif. 91103. 1 T. Kasami, “An upper bound on k/n for affine-invariant codes with fixed d/n, “IEEE Trans. Information Theory (Correspondence), vol. IT-1.5, pp. 174-176, January 1969. p, P, = affine group of order n (n 1) if n = 2” some m.