Local-Testability and Self-Correctability of q-ary Sparse Linear Codes

TL;DR

This paper demonstrates that q-ary sparse codes with small bias are both self-correctable and locally testable, extending previous binary code results using properties of q-ary Krawtchouk polynomials and the McWilliams identity.

Contribution

It generalizes the local testability and self-correctability results from binary to q-ary sparse codes with small bias, employing advanced polynomial and weight distribution techniques.

Findings

Proves q-ary sparse codes with small bias are self-correctable.

Establishes local testability of q-ary sparse codes.

Provides bounds on error probability using polynomial properties.

Abstract

We prove that q-ary sparse codes with small bias are self-correctable and locally testable. We generalize a result of Kaufman and Sudan that proves the local testability and correctability of binary sparse codes with small bias. We use properties of q-ary Krawtchouk polynomials and the McWilliams identity -that relates the weight distribution of a code to the weight distribution of its dual- to derive bounds on the error probability of the randomized tester and self-corrector we are analyzing.