Optimal Additive Quaternary Codes of Low Dimension
2021
An additive quaternary
$[n,k,d]$
-code (length
$n$
, quaternary dimension
$k$
, minimum distance
$d$
) is a
$2k$
-dimensional
$\mathbb {F}_{2}$
-vector space of
$n$
-tuples with entries in
$\mathbb {F}_{2}\oplus \mathbb {F} _{2}$
(the 2-dimensional vector space over
$\mathbb {F}_{2}$
) with minimum Hamming distance
$d$
. We determine the optimal parameters of additive quaternary codes of dimension
$k\leq 3$
. The most challenging case is dimension
$k=2.5$
. We prove that an additive quaternary
$[n,2.5,d]$
-code where
$d< n-1$
exists if and only if
$3(n-d)\geq \lceil d/2\rceil +\lceil d/4\rceil +\lceil d/8\rceil $
. 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)\geq \lceil e/2\rceil +\lceil e/4\rceil +\lceil e/8\rceil $
is satisfied.
- Correction
- Source
- Cite
- Save
- Machine Reading By IdeaReader
9
References
0
Citations
NaN
KQI