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 2k-dimensional \mathbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -vector space of n-tuples with entries in \mathbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ⊕\mathbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (the 2-dimensional vector space over \mathbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ) 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 where d <; n-1 exists if and only if 3(n-d) ≥ d/2+d/4+d/8 articular, 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] <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -code for e <; m-1 exists if and only if the Griesmer bound 3(m-e) ≥ e/2+e/4+e/8 is satisfied.