Error-correction of linear codes via colon ideals

TL;DR

This paper introduces a novel algebraic approach to error correction in linear codes by using colon ideals and primary decomposition of homogeneous ideals, providing a new method to identify and correct errors efficiently.

Contribution

It presents a new algebraic technique involving colon ideals and primary decomposition to determine error locations in linear codes, improving error correction methods.

Findings

Error correction can be modeled as finding minimum weight codewords in new linear codes.

Primary decomposition of saturated ideals can identify error positions.

A method to compute the ideal using coloning with a variable power is established.

Abstract

We show that errors in data transmitted through linear codes can be thought of as codewords of minimum weight of new linear codes. To determine errors we can then use methods specific to finding such special codewords. One of these methods consists of finding the primary decomposition of the saturation of a certain homogeneous ideal. When good words (i.e. vectors with a unique nearest neighbor) are error-corrected, the saturated ideal is just the prime ideal of a point (so the primary decomposition is superfluously determined); we show that this ideal can be computed by coloning the original homogeneous ideal with a power of a certain variable. We then determine the smallest such power for any linear code.