Optimal Additive Quaternary Codes of Low Dimension

Abstract: An additive quaternary

[n, k, d]

-code (length

n,

quaternary dimension

k,

minimum distance

d

) is a

2 k

-dimensional F_2-vector space of

n

-tuples with entries in

Z_{2} i m e s Z_{2}

(the

2

-dimensional vector space over F_2) with minimum Hamming distance

d .

We determine the optimal parameters of additive quaternary codes of dimension

k l e q 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) g e q l c e i l d / 2 c e i l + l c e i l d / 4 c e i l + l c e i l d / 8 c e i l

. 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

[3 m, 5, 2 e]_{2}

-code for

e < m - 1

exists if and only if the Griesmer bound

3 (m - e) g e q l c e i l e / 2 c e i l + l c e i l e / 4 c e i l + l c e i l e / 8 c e i l

is satisfied.