Fast Decoding of Explicit Almost Optimal ε-Balanced q-Ary Codes And Fast Approximation of Expanding k-CSPs

Good codes over an alphabet of constant size q can approach but not surpass distance 1 − 1 /q . This makes the use of q -ary codes a necessity in some applications, and much work has been devoted to the case of constant alphabet q . In the large distance regime, namely, distance 1 − 1 /q − ε for small ε > 0, the Gilbert–Varshamov (GV) bound asserts that rate Ω q ( ε 2 ) is achievable whereas the q -ary MRRW bound gives a rate upper bound of O q ( ε 2 log(1 /ε )). In this sense, the GV bound is almost optimal in this regime. Prior to this work there was no known explicit and efficiently decodable q -ary codes near the GV bound, in this large distance regime, for any constant q ≥ 3. We design an e O ε,q ( N ) time decoder for explicit (expander based) families of linear codes C N,q,ε ⊆ F Nq of distance (1 − 1 /q )(1 − ε ) and rate Ω q ( ε 2+ o (1) ), for any desired ε > 0 and any constant prime q , namely, almost optimal in this regime. These codes are ε -balanced,i.e., for every non-zero codeword, the frequency of each symbol lies in the interval [1 /q − ε, 1 /q + ε ]. A key ingredient of the q -ary decoder is a new near-linear time approximation algorithm for linear equations ( k -LIN) over Z q on expanding hypergraphs, in particular, those naturally arising in the decoding of these codes. We also investigate k -CSPs on expanding hypergraphs in more generality. We show that special trade-offs available for k -LIN over Z q hold for linear equations over a finite group. To handle general finite groups, we design a new matrix version of weak regularity for expanding hypergraphs. We also obtain a near-linear time approximation algorithm for general expanding k -CSPs over q -ary alphabet. This later algorithm runs in time e O k,q ( m + n ), where m is the number of constraints and n is the number of variables. This improves the previous best running time of O ( n Θ k,q (1) ) by a Sum-of-Squares based algorithm of [AJT, 2019] (in the expanding regular case). We obtain our results by generalizing the framework of [JST, 2021] based on weak regularity decomposition for expanding hypergraphs. This framework was originally designed for binary k -XOR with the goal of providing near-linear time decoder for explicit binary codes, near the GV bound, from the breakthrough work of Ta-Shma [STOC, 2017]. The explicit families of codes over prime F q are based on suitable instatiations of the Jalan–Moshkovitz (Abelian) generalization of Ta-Shma’s distance amplification procedure.