The Size of the Intersection of q-ary Hamming Balls

Abstract

The interest in studying the size of the intersection of multiple q-ary Hamming balls has grown due to the recent advances in DNA-based data storage systems. We present an exact formula for the cardinality of the intersection of s Hamming balls of varying radii over a q-ary alphabet. It is known that the distances between the center points of the Hamming balls are not enough, in general, to determine the size of the intersection. Based on our formula, we are able to find more refined structural properties of the center points for determining the exact size of the intersection. Moreover, we also analyze the size of the intersection of Hamming balls with equal radii t for sufficiently large n when t and q are constants. When s=3, we give the necessary and sufficient conditions (for all q≥ 2, q≠ 6 and sufficiently large n) to obtain the maximum size of the intersection when the centers of the Hamming balls have a given minimum distance and demonstrate how to compute it using our general formula.

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BibTeX RIS

BibTeX

@misc{junnila2026size,
  title = {{The Size of the Intersection of $q$-ary Hamming Balls}},
  author = {Ville Junnila and Tero Laihonen and Tuomo Lehtilä and Pavan Padavu Devaraj},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.09158},
  eprint = {2606.09158},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.09158},
  note = {CC BY-SA 4.0},
}

RIS

TY  - UNPB
TI  - The Size of the Intersection of q-ary Hamming Balls
AU  - Ville Junnila
AU  - Tero Laihonen
AU  - Tuomo Lehtilä
AU  - Pavan Padavu Devaraj
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2606.09158
UR  - https://arxiv.org/abs/2606.09158
AB  - The interest in studying the size of the intersection of multiple q-ary Hamming balls has grown due to the recent advances in DNA-based data storage systems. We present an exact formula for the cardinality of the intersection of s Hamming balls of varying radii over a q-ary alphabet. It is known that the distances between the center points of the Hamming balls are not enough, in general, to determine the size of the intersection. Based on our formula, we are able to find more refined structural properties of the center points for determining the exact size of the intersection. Moreover, we also analyze the size of the intersection of Hamming balls with equal radii t for sufficiently large n when t and q are constants. When s=3, we give the necessary and sufficient conditions (for all q≥ 2, q≠ 6 and sufficiently large n) to obtain the maximum size of the intersection when the centers of the Hamming balls have a given minimum distance and demonstrate how to compute it using our general formula.
ER  -

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