An additive quaternary [n, k, d]-code (length n, quaternary dimension k, minimum distance d) is a 2kdimensional F2-vector space of n-Tuples with entries in F2-F2 (the 2-dimensional vector space over F2) with minimum Hamming distance d. We determine the optimal parameters of additive quaternary codes of dimension k-3. The most challenging case is dimension k = 2.5. We prove that an additive quaternary [n, 2.5, d]-code whered n-1 exists if and only if 3(n-d) d/2+d/4+d/8. In particular, we construct new optimal 2.5-dimensional additive quaternary codes. As a by-product, we give a direct proof for the fact that a binary linear [3m, 5, 2e]2-code for e m-1 exists if and only if the Griesmer bound 3(m-e) e/2+e/4+e/8 is satisfied. .
Optimal Additive Quaternary Codes of Low Dimension
Bierbrauer J.;Marcugini S.;Pambianco F.
2021
Abstract
An additive quaternary [n, k, d]-code (length n, quaternary dimension k, minimum distance d) is a 2kdimensional F2-vector space of n-Tuples with entries in F2-F2 (the 2-dimensional vector space over F2) with minimum Hamming distance d. We determine the optimal parameters of additive quaternary codes of dimension k-3. The most challenging case is dimension k = 2.5. We prove that an additive quaternary [n, 2.5, d]-code whered n-1 exists if and only if 3(n-d) d/2+d/4+d/8. In particular, we construct new optimal 2.5-dimensional additive quaternary codes. As a by-product, we give a direct proof for the fact that a binary linear [3m, 5, 2e]2-code for e m-1 exists if and only if the Griesmer bound 3(m-e) e/2+e/4+e/8 is satisfied. .I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
