A Complete Characterization of q-Ary $2$-Designs in Doubly-Extended Reed-Solomon Codes

Abstract

Let 𝒞_k be the doubly-extended Reed--Solomon code with parameters [q+1, k, q−k+2]_q. We give a complete characterization of when its codewords of a fixed weight form q-ary 2-designs. The main ingredient is an exact criterion showing that the number of r-subsets of 𝔽_q^* with a prescribed product is independent of that product if and only if gcd(r, q−1)=1. Applying this criterion to the minimum-weight codewords of 𝒞_k, we prove that they form a q-ary 2-design if and only if gcd(k−1, q−1)=1. We further establish a propagation result for MDS codes showing that the q-ary 2-design property at minimum weight implies the same property at every nonzero weight. Consequently, 𝒞_k holds q-ary 2-designs at all nonzero weights if and only if gcd(k−1, q−1)=1.

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BibTeX RIS

BibTeX

@misc{wu2026complete,
  title = {{A Complete Characterization of $q$-Ary $2$-Designs in Doubly-Extended Reed-Solomon Codes}},
  author = {Xinghao Wu and Junling Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05861},
  eprint = {2611.05861},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05861},
}

RIS

TY  - UNPB
TI  - A Complete Characterization of q-Ary $2$-Designs in Doubly-Extended Reed-Solomon Codes
AU  - Xinghao Wu
AU  - Junling Zhou
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.05861
UR  - https://arxiv.org/abs/2611.05861
AB  - Let 𝒞_k be the doubly-extended Reed--Solomon code with parameters [q+1, k, q−k+2]_q. We give a complete characterization of when its codewords of a fixed weight form q-ary 2-designs. The main ingredient is an exact criterion showing that the number of r-subsets of 𝔽_q^* with a prescribed product is independent of that product if and only if gcd(r, q−1)=1. Applying this criterion to the minimum-weight codewords of 𝒞_k, we prove that they form a q-ary 2-design if and only if gcd(k−1, q−1)=1. We further establish a propagation result for MDS codes showing that the q-ary 2-design property at minimum weight implies the same property at every nonzero weight. Consequently, 𝒞_k holds q-ary 2-designs at all nonzero weights if and only if gcd(k−1, q−1)=1.
ER  -

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