On Computation of Error Locations and Values in Hermitian Codes

TL;DR

This paper introduces a new semi-erasure decoding technique for Hermitian codes that simplifies error location and value computation, reducing complexity by leveraging univariate polynomials and adapting Reed-Solomon decoding methods.

Contribution

It proposes a semi-erasure decoding framework for Hermitian codes, enabling error location and value computation with reduced complexity using univariate polynomials.

Findings

Error locations found via univariate polynomial evaluation over q^2 points.

Forney's formula applies directly for error value computation in Hermitian codes.

Decoding Hermitian codes can always be performed using semi-erasure decoding.

Abstract

We obtain a technique to reduce the computational complexity associated with decoding of Hermitian codes. In particular, we propose a method to compute the error locations and values using an uni-variate error locator and an uni-variate error evaluator polynomial. To achieve this, we introduce the notion of Semi-Erasure Decoding of Hermitian codes and prove that decoding of Hermitian codes can always be performed using semi-erasure decoding. The central results are: * Searching for error locations require evaluating an univariate error locator polynomial over $q^2$ points as in Chien search for Reed-Solomon codes. * Forney's formula for error value computation in Reed-Solomon codes can directly be applied to compute the error values in Hermitian codes. The approach develops from the idea that transmitting a modified form of the information may be more efficient that the information itself.