Abstract
The definition of the Feng-Rao designed minimum distance, first introduced into algebraic geometry codes, has been extended to the case of general linear codes by Miura. In Miura’s definition the Feng-Rao designed minimum distance dFR is determined by an ordered basis related to a given linear code over a finite field Fq Matsumoto gave a generalized definition d*FR of dFR with three ordered bases. We have Miura’s definition if three ordered bases are same in Matsumoto’s definition. In this Chapter, firstly, it is shown that the Feng-Rao designed minimum distance of binary linear codes cannot take an odd value except one if we use Miura’s definition. Secondly, from some properties and some numerical examples of Matsumoto’s d*FR we conjecture that Matsumoto’s generalization is not so effective for binary linear codes compared with Miura’s definition. Thirdly, the Type I ordered basis is introduced for computing dFR of cyclic codes. It is shown that the Type I ordered basis is worst choice in case of nonbinary cyclic codes, although it is not so bad in case of binary (n, 1), (n, 2) and (n, n-1) cyclic codes.Key wordsbinary linear codeFeng-Raodesigned minimum distanceordered basisMiura’s definitionMatsumoto’s definition
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