Perfect codes as exact minimizers of quadratic discrepancy in q-ary Hamming spaces
Abstract
Total quadratic ball discrepancy measures the deviation of codeword counts from uniformity over all Hamming balls. We prove that, whenever a Hamming space and cardinality admit a perfect code, the discrepancy minimizers among all subsets of that cardinality are precisely the perfect codes. This extends Barg's binary minimizing result and characterizes equality over arbitrary finite alphabets. At the necessary parameter sets for correcting one or two errors, we give explicit lower bounds whose attainment is equivalent to perfect tiling, even when existence is unresolved. For alphabets of size at least four, the numerical values follow from known universal energy bounds; we establish their discrepancy normalization and the attainment criterion. The ternary discrepancy potential falls outside the completely monotonic regime of those bounds, and separate exact certificates settle the ternary Hamming and Golay cases.
Related
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- Quaternary Preparata codes are quasi-perfect covering codes in the Hamming metric
- Ternary Perfect Forms Over Imaginary Quadratic Field
- The Asymptotic Number of Anticodes in Hamming Spaces
- An overlooked weakening of perfect normality: Perfect regularity in spaces and locales
BibTeX
@misc{zabokritskiy2026perfect,
title = {{Perfect codes as exact minimizers of quadratic discrepancy in q-ary Hamming spaces}},
author = {Aryeh Lev Zabokritskiy},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.01134},
eprint = {2608.01134},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.01134},
}
RIS
TY - UNPB
TI - Perfect codes as exact minimizers of quadratic discrepancy in q-ary Hamming spaces
AU - Aryeh Lev Zabokritskiy
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2608.01134
UR - https://arxiv.org/abs/2608.01134
AB - Total quadratic ball discrepancy measures the deviation of codeword counts from uniformity over all Hamming balls. We prove that, whenever a Hamming space and cardinality admit a perfect code, the discrepancy minimizers among all subsets of that cardinality are precisely the perfect codes. This extends Barg's binary minimizing result and characterizes equality over arbitrary finite alphabets. At the necessary parameter sets for correcting one or two errors, we give explicit lower bounds whose attainment is equivalent to perfect tiling, even when existence is unresolved. For alphabets of size at least four, the numerical values follow from known universal energy bounds; we establish their discrepancy normalization and the attainment criterion. The ternary discrepancy potential falls outside the completely monotonic regime of those bounds, and separate exact certificates settle the ternary Hamming and Golay cases.
ER -