Quaternary Preparata codes are quasi-perfect covering codes in the Hamming metric

Abstract

The quaternary Preparata codes of Hammons, Kumar, Calderbank, Sloane and Sol\'e are $\mathbb{Z}_4$-linear codes of length $2^m$ with $4^{2^m-m-1}$ codewords. They are usually studied through their binary Gray images, that is, in the Lee metric. We consider them instead as codes over a 4-letter alphabet with the Hamming metric. We show that for odd $m$ they have covering radius 2 and minimum distance 4, so they are quasi-perfect. For even $m$ the same construction has covering radius 3. The proof is a short computation with Teichm\"uller representatives in the Galois ring $\mathrm{GR}(4,m)$, and it also gives the coset weight distribution. The covering density of these codes tends to $9/8$. The best previously known infinite families of quaternary codes with covering radius 2 have covering density about 1.5. Consequently $K_4(n,2)\le 4^{n-m-1}$ for all $n\ge 2^m$ and odd $m$, which improves the best available bounds for a range of lengths above each $2^m$. For $m=3$ the code is the octacode, and it gives $K_4(8,2)\le 256$; the previous bound was 352, from 2005, and the best lower bound is 251. We also give a quaternary code showing $K_4(6,2)\le 48$, where the previous bound was 52, together with consequences for larger alphabets.