Abstract
Some definitions, theorems and lemmas related to optimal codes on binary field have been proposed in this paper. Optimal codes with different dimensions have been gotten by directly constructing, combinating and deleting. Relevant proofs have also been given. We work on how to gain optimal codes when 2≤k≤5. Generator matrices of these optimal codes can be constructed, and codes that reaches the Griesmer bound can be furthermore determined. Method adopted in this paper is based on check matrices and designed to construct optimal codes which have low dimensions, it provides a theoretical basis for codes with higher dimensions.
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