Applications of Grobner Bases to Linear Codes.

Put A = F , [ z i , . . . ,Za], and le t / be an ideal o f .4. Let be a ll the Fç-rational points o f V { I ) . Define a map ip : A / 1 — F " by (p{f) = ( / ( P i ) , . . . , / (P „ ) ) , where / is any preimage o f / under the canonical map from A to A / / . Let { / , | i E N } be a basis o f A / / as an F ,-vector space. Define the affine variety code C and its dual C'*" by ^ ~ V^((/l7 • • • ) fm))'i is the orthogonal complement o f C w ith respect to the usual inner product in F ". We show tha t any linear code can be expressed as an affine variety code. When a code C is represented as an affine variety code, problems o f de­ coding and find ing the m inim um distance o f C may be expressed as questions about polynom ial ideals. Using the theory o f Grobner bases, along w ith computer programs th a t cal­ culate Grobner bases, we show how to decode and find the m inim um distance o f any linear code.