Error-correction of linear codes via colon ideals

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.

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