Quaternary Preparata codes are quasi-perfect covering codes in the Hamming metric
Abstract
The quaternary Preparata codes of Hammons, Kumar, Calderbank, Sloane and Solé are ℤ4-linear codes of length 2m with 42m−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 Teichmüller representatives in the Galois ring 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 K4(n,2)≤ 4n−m−1 for all n≥ 2m and odd m, which improves the best available bounds for a range of lengths above each 2m. For m=3 the code is the octacode, and it gives K4(8,2)≤ 256; the previous bound was 352, from 2005, and the best lower bound is 251. We also give a quaternary code showing K4(6,2)≤ 48, where the previous bound was 52, together with consequences for larger alphabets.
Related
- Perfect codes as exact minimizers of quadratic discrepancy in q-ary Hamming spaces
- Covering Bridgeless Claw-Free Cubic Graphs with Perfect Matchings
- Weak k-metric dimension of Hamming graphs: rectangular products and near-maximum parameters
- Besicovitch's covering theorem in the parabolic metric
- Mordukhovich Derivatives and Covering Constant for Set-Valued Metric Projection in General Banach Spaces
BibTeX
@misc{maharaj2026quaternary,
title = {{Quaternary Preparata codes are quasi-perfect covering codes in the Hamming metric}},
author = {Tamal Maharaj},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03760},
eprint = {2610.03760},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03760},
}
RIS
TY - UNPB
TI - Quaternary Preparata codes are quasi-perfect covering codes in the Hamming metric
AU - Tamal Maharaj
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.03760
UR - https://arxiv.org/abs/2610.03760
AB - The quaternary Preparata codes of Hammons, Kumar, Calderbank, Sloane and Solé are 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 Teichmüller representatives in the Galois ring 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)≤ 4^n−m−1 for all n≥ 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)≤ 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)≤ 48, where the previous bound was 52, together with consequences for larger alphabets.
ER -