Distance and Symmetry: Two Pillars of a Good Code

Abstract

This thesis explores the benefits of distance and symmetry in the design of error correcting codes. Our main measure of distance will be the minimum distance of a code, i.e., the smallest number of coordinates any two codewords may differ in. Our main criterion for symmetry will be transitivity, i.e., the requirement that any two coordinates be interchangeable. We will also consider generalizations of these two notions, for instance double transitivity (the requirement that any two pairs of coordinates be interchangeable) and generalized distances (the minimum number of nontrivial coordinates in any subcode of a certain size). We argue that codes with large distances and high symmetry present desirable properties for communication on noisy channels. Concretely, we prove the following results: 1) Any linear code that achieves list decoding capacity and has superconstant minimum distance also achieves capacity on the symmetric channel. 2) For any linear code C with large enough generalized distances, the bit-decoding and block-decoding thresholds of C on the erasure channel are asymptotically equal. 3) Any transitive linear code $C\subseteq\F_q^N$ contains at most $q^{h_q(\alpha) \cdot\textnormal{dim }C }$ codewords of weight $\alpha N$. This upper bound is tight, as evidenced by repetition codes. 4) For every doubly transitive code C, there is a range of noises for which C achieves the information-theoretic optimal trade-off between rate and list decoding size. 5) The canonical example of linear codes with large distances and high symmetries is the family of Reed-Muller codes. We show that for appropriate choices of Reed-Muller codes $C_1$, $C_2$ and $C_3$, the tensor code $C_1\otimes C_2\otimes C_3$ achieves capacity on the symmetric channel with quasilinear decoding time.