Abstract
In this paper, the 3-rank of the incidence matrices of 2-designs supported by the minimum weight codewords in a family of ternary linear codes considered in Ding and Li (Discret Math 340(10):2415–2431, 2017) are computed. A lower bound on the minimum distance of the ternary codes spanned by the incidence matrices of these designs is derived, and it is proved that the codes are subcodes of the 4th order generalized Reed–Muller codes.
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Notes
- 1.
More generally, a code \({\mathsf {C}}\) holds (or supports) a t-\((v,k,\lambda )\) design \({{\mathbb {D}}}\) if every block of \({{\mathbb {D}}}\) is the support of some codeword of \({\mathsf {C}}\) [23].
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The authors are grateful to the reviewers and the Editors for their comments and suggestions that improved the presentation of this paper.
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C. Ding’s research was supported by the Hong Kong Research Grants Council, Proj. No. 16300418. C. Tang was supported by National Natural Science Foundation of China (Grant No. 11871058) and China West Normal University (14E013, CXTD2014-4 and the Meritocracy Research Funds).
Communicated by J. D. Key.
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Ding, C., Tang, C. & Tonchev, V.D. Linear codes of 2-designs associated with subcodes of the ternary generalized Reed–Muller codes. Des. Codes Cryptogr. 88, 625–641 (2020). https://doi.org/10.1007/s10623-019-00701-1
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Keywords
- Cyclic code
- Linear code
- Reed–Muller code
- t-design
Mathematics Subject Classification
- 94B05
- 94B15
- 05B05