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Further classifications of codes meeting the Griesmer bound (Corresp.)

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For any (n, k, d) binary linear code, the Griesmer bound says that n \geq \sum _{i=0}^{k-1} \lceil d/2^{i} \rceil , where \lceil x \rceil denotes the smallest integer \geq x . We consider codes meeting the Griesmer bound with equality. These codes have parameters \left( s(2^{k} - 1) - \sum _{i=1}^{p} (2^{u_{i}} - 1), k, s2^{k-1} - \sum _{i=1}^{p} 2^{u_{i} -1} \right) , where k > u_{1} > \cdots > u_{p} \geq 1 . We characterize all such codes when p = 2 or u_{i-1}-u_{i} \geq 2 for 2 \leq i \leq p .

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Information Theory, IEEE Transactions on  (Volume:30 ,  Issue: 2 )