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Geometric orthogonal codes and geometrical difference packings

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Abstract

Motivated by their application in the synthesis of DNA nanomaterials, geometric orthogonal codes (briefly GOCs) were introduced recently. We establish the equivalence between \((n_1\times n_2,w,1)\)-GOCs and a certain type of combinatorial configurations, called \((n_1\times n_2,w,1)\)-geometrical difference packings. Based on this relationship, upper bounds for the number of codewords are presented and recursive constructions are provided. As a consequence, the number of codewords in an optimal \((n_1\times n_2,3,1)\)-GOC is determined for any positive integers \(n_1\) and \(n_2\).

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Acknowledgements

The authors would like to thank the anonymous reviewers and Prof. Marco Buratti, the Coordinating Editor, for their helpful comments and valuable suggestions to improve the quality and readability of this paper.

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Authors and Affiliations

  1. School of Intelligence Policing, China People’s Police University, Langfang, 065000, People’s Republic of China

    Lidong Wang

  2. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, People’s Republic of China

    Lulu Cai & Zihong Tian

  3. School of Mathematics and Statistics, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Tao Feng

  4. School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, People’s Republic of China

    Xiaomiao Wang

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  1. Lidong Wang
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  2. Lulu Cai
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  3. Tao Feng
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  4. Zihong Tian
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  5. Xiaomiao Wang
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Correspondence to Xiaomiao Wang.

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Communicated by M. Buratti.

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Supported by NSFC under Grant 11901210, and the Key R &D Program of Guangdong Province under Grant 2020B030304002 (L. Wang), NSFC under Grant 11871095, 12171028 (T. Feng), NSFC under Grant 11871019 (Z. Tian), Zhejiang Provincial Natural Science Foundation of China under Grant LY21A010005, NSFC under Grant 11771227, the Fundamental Research Funds for the Provincial Universities of Zhejiang under Grant SJLY2020008, and Natural Science Foundation of Ningbo under Grant 202003N4141 (X. Wang).

Appendices

Appendix: Optimal \((n_1\times n_2,3,1)\)-GDPs

Mentioned in Lemma 6.10:

$$\begin{aligned} \begin{array}{llllllll} \cdot &{}(n_1,n_2)=(5,17):\\ &{}\{(0,0),(2,1),(2,4)\}, &{}\{(0,0),(1,0),(2,7)\}, &{}\{(0,0),(0,2),(1,1)\}, &{}\{(0,0),(1,2),(2,0)\},\\ {} &{} \{(0,0),(0,7),(1,3)\}, &{}\{(0,6),(2,0),(2,5)\}, &{}\{(0,8),(1,0),(2,4)\}, &{}\{(0,0),(0,8),(2,3)\},\\ {} &{} \{(0,0),(0,6),(2,8)\}, &{}\{(0,3),(0,7),(2,0)\}, &{}\{(0,1),(2,6),(2,7)\}, &{}\{(0,8),(1,3),(2,0)\},\\ {} &{} \{(0,8),(1,1),(2,6)\}.\\ \cdot &{}(n_1,n_2)=(11,17):\\ {} &{} \{(0,0),(1,0),(5,8)\}, &{}\{(0,0),(2,0),(5,7)\}, &{}\{(0,0),(3,0),(5,6)\}, &{}\{(0,0),(5,0),(0,8)\},\\ {} &{} \{(0,0),(4,0),(4,7)\}, &{}\{(0,0),(2,1),(5,5)\}, &{}\{(0,0),(4,1),(3,8)\}, &{}\{(0,0),(3,1),(4,6)\},\\ {} &{} \{(0,0),(5,1),(5,4)\}, &{}\{(0,0),(1,2),(2,8)\}, &{}\{(0,0),(0,2),(4,5)\}, &{}\{(0,0),(5,3),(0,4)\},\\ {} &{} \{(0,0),(4,2),(3,6)\}, &{}\{(0,0),(3,2),(2,7)\}, &{}\{(0,0),(5,2),(1,8)\}, &{}\{(0,0),(4,4),(0,5)\},\\ {} &{} \{(0,0),(3,5),(0,6)\}, &{}\{(1,0),(2,7),(0,8)\}, &{}\{(2,0),(4,5),(0,7)\}, &{}\{(3,0),(5,3),(0,5)\},\\ {} &{} \{(2,0),(5,3),(0,6)\}, &{}\{(2,0),(4,4),(0,8)\}, &{}\{(3,0),(4,4),(0,7)\}, &{}\{(4,0),(5,3),(0,7)\},\\ {} &{} \{(4,0),(1,2),(0,8)\}, &{}\{(5,0),(3,2),(0,5)\}, &{}\{(3,0),(2,3),(0,8)\}, &{}\{(3,0),(2,2),(0,6)\},\\ {} &{} \{(5,0),(4,1),(0,6)\}, &{}\{(5,0),(3,3),(0,7)\}. \end{array} \end{aligned}$$

Mentioned in Lemma 6.11:

$$\begin{aligned} \begin{array}{lllllll} \cdot &{}(n_1,n_2)=(5,13):\\ {} &{} \{(0,0),(2,6),(1,1)\}, &{}\{(0,0),(1,6),(2,5)\}, &{}\{(0,0),(2,3),(2,2)\}, &{}\{(0,0),(2,0),(0,2)\},\\ {} &{} \{(0,2),(2,1),(0,5)\}, &{}\{(0,0),(2,1),(0,6)\}, &{}\{(0,0),(1,0),(1,4)\}, &{}\{(0,6),(2,0),(1,2)\},\\ {} &{} \{(0,3),(2,0),(1,6)\}, &{}\{(0,5),(1,2),(0,0)\}.\\ \cdot &{}(n_1,n_2)=(11,13):\\ {} &{} \{(0,0),(5,2),(3,5)\}, &{}\{(0,0),(0,2),(3,2)\}, &{}\{(0,0),(1,1),(1,5)\}, &{}\{(0,0),(5,3),(2,6)\},\\ {} &{} \{(0,0),(5,5),(0,6)\}, &{}\{(0,0),(1,4),(5,6)\}, &{}\{(0,0),(4,0),(3,3)\}, &{}\{(0,0),(0,1),(4,4)\},\\ {} &{} \{(0,0),(5,1),(5,4)\}, &{}\{(0,0),(3,1),(0,5)\}, &{}\{(0,0),(5,0),(4,5)\}, &{}\{(0,0),(2,1),(4,6)\},\\ {} &{} \{(0,0),(1,2),(3,4)\}, &{}\{(0,0),(1,6),(3,6)\}, &{}\{(5,0),(3,2),(0,4)\}, &{}\{(5,0),(2,1),(3,4)\},\\ {} &{} \{(5,0),(1,4),(0,6)\}, &{}\{(5,0),(4,4),(2,5)\}, &{}\{(5,0),(0,5),(4,6)\}, &{}\{(5,0),(1,3),(3,6)\},\\ {} &{} \{(5,0),(1,1),(3,5)\}, &{}\{(5,0),(1,6),(2,6)\}, &{}\{(5,0),(4,1),(0,3)\}.\\ \cdot &{}(n_1,n_2)=(5,19):\\ {} &{} \{(0,0),(2,3),(1,9)\}, &{}\{(0,0),(1,1),(2,4)\}, &{}\{(1,0),(1,7),(0,9)\}, &{}\{(1,0),(1,2),(2,2)\},\\ {} &{} \{(2,0),(2,4),(0,7)\}, &{}\{(2,0),(0,6),(0,9)\}, &{}\{(2,0),(0,5),(2,5)\}, &{}\{(2,0),(0,1),(1,8)\},\\ {} &{} \{(2,0),(0,4),(2,9)\}, &{}\{(0,1),(2,8),(2,9)\}, &{}\{(0,1),(2,2),(1,9)\}, &{}\{(0,1),(2,3),(1,7)\},\\ {} &{} \{(0,1),(1,6),(0,7)\}, &{}\{(0,1),(2,7),(0,9)\}, &{}\{(2,1),(1,4),(0,9)\}.\\ \cdot &{}(n_1,n_2)=(11,19):\\ {} &{} \{(0,0),(1,0),(5,9)\}, &{}\{(0,0),(5,0),(0,9)\}, &{}\{(0,0),(2,0),(5,8)\}, &{}\{(1,0),(5,0),(0,8)\},\\ {} &{} \{(0,0),(3,0),(5,7)\}, &{}\{(0,0),(5,1),(0,8)\}, &{}\{(0,0),(0,1),(5,6)\}, &{}\{(0,0),(1,1),(4,7)\},\\ {} &{} \{(0,0),(0,2),(3,9)\}, &{}\{(0,0),(3,1),(4,8)\}, &{}\{(0,0),(4,1),(4,6)\}, &{}\{(0,0),(1,2),(5,4)\},\\ {} &{} \{(0,0),(2,2),(4,5)\}, &{}\{(0,0),(3,2),(2,9)\}, &{}\{(0,0),(5,3),(4,4)\}, &{}\{(0,0),(5,2),(1,9)\},\\ {} &{} \{(0,0),(0,3),(2,8)\}, &{}\{(0,0),(3,5),(0,6)\}, &{}\{(0,0),(3,3),(1,8)\}, &{}\{(2,0),(5,4),(0,8)\},\\ {} &{} \{(1,0),(2,3),(0,9)\}, &{}\{(3,0),(4,4),(0,8)\}, &{}\{(3,0),(5,4),(0,6)\}, &{}\{(2,0),(4,6),(0,9)\},\\ {} &{} \{(4,0),(5,5),(0,6)\}, &{}\{(1,0),(5,3),(0,6)\}, &{}\{(3,0),(4,6),(0,7)\}, &{}\{(3,0),(0,2),(0,9)\},\\ {} &{} \{(4,0),(2,1),(0,5)\}, &{}\{(2,0),(0,3),(0,7)\}, &{}\{(4,0),(1,3),(0,8)\}, &{}\{(5,0),(1,2),(0,5)\},\\ {} &{} \{(4,0),(3,4),(0,9)\}, &{}\{(5,0),(3,2),(0,6)\}. \end{array} \end{aligned}$$

Mentioned in Lemma 6.12:

$$\begin{aligned} \begin{array}{llllll} \cdot &{} (n_1,n_2)=(13,19):\\ {} &{} \{(0,0),(1,0),(6,9)\}, &{}\{(0,0),(2,0),(6,8)\}, &{}\{(0,0),(6,0),(0,9)\}, &{}\{(0,0),(3,0),(6,7)\},\\ {} &{} \{(1,0),(6,0),(0,8)\}, &{}\{(0,0),(4,0),(6,6)\}, &{}\{(0,0),(3,1),(5,8)\}, &{}\{(0,0),(2,1),(4,9)\},\\ {} &{} \{(0,0),(4,1),(6,5)\}, &{}\{(0,0),(5,1),(5,7)\}, &{}\{(0,0),(6,1),(3,9)\}, &{}\{(0,0),(0,2),(5,6)\},\\ {} &{} \{(0,0),(1,2),(6,4)\}, &{}\{(0,0),(2,2),(5,5)\}, &{}\{(0,0),(3,2),(4,7)\}, &{}\{(0,0),(4,2),(2,9)\},\\ {} &{} \{(0,0),(6,2),(4,6)\}, &{}\{(0,0),(0,3),(1,9)\}, &{}\{(0,0),(1,3),(3,8)\}, &{}\{(0,0),(6,3),(0,4)\},\\ {} &{} \{(1,0),(6,3),(0,5)\}, &{}\{(0,0),(4,3),(1,8)\}, &{}\{(0,0),(4,5),(0,8)\}, &{}\{(1,0),(0,2),(0,9)\},\\ {} &{} \{(1,0),(4,6),(0,7)\}, &{}\{(2,0),(5,5),(0,9)\}, &{}\{(2,0),(6,4),(0,8)\}, &{}\{(3,0),(4,7),(0,9)\},\\ {} &{} \{(3,0),(6,4),(0,7)\}, &{}\{(5,0),(0,1),(0,6)\}, &{}\{(6,0),(1,2),(0,5)\}, &{}\{(4,0),(1,3),(0,9)\},\\ {} &{} \{(4,0),(5,4),(0,7)\}, &{}\{(5,0),(2,2),(0,8)\}, &{}\{(6,0),(3,1),(0,7)\}, &{}\{(5,0),(4,1),(0,7)\},\\ {} &{} \{(5,0),(3,1),(0,5)\}, &{}\{(5,0),(4,4),(0,9)\}, &{}\{(6,0),(4,2),(0,6)\}, &{}\{(4,0),(2,3),(0,8)\}. \end{array} \end{aligned}$$

Mentioned in Lemma 6.13:

$$\begin{aligned} \begin{array}{llllll} \cdot &{} (n_1,n_2)=(13,17):\\ {} &{} \{(0,0),(1,1),(6,8)\},&{}\{(0,8),(1,7),(6,0)\},&{}\{(0,0),(1,2),(6,7)\},&{}\{(0,7),(1,5),(6,0)\},\\ {} &{} \{(0,0),(1,8),(6,5)\},&{}\{(0,8),(1,0),(6,3)\},&{}\{(0,0),(2,1),(6,6)\},&{}\{(0,6),(2,5),(6,0)\},\\ {} &{} \{(0,0),(2,8),(6,2)\},&{}\{(0,8),(2,0),(6,6)\},&{}\{(0,0),(2,6),(5,8)\},&{}\{(0,8),(2,2),(5,0)\},\\ {} &{} \{(0,0),(3,7),(5,2)\},&{}\{(0,7),(3,0),(5,5)\},&{}\{(0,0),(3,4),(5,1)\},&{}\{(0,4),(3,0),(5,3)\},\\ {} &{} \{(0,0),(4,1),(5,4)\},&{}\{(0,4),(4,3),(5,0)\},&{}\{(0,0),(1,4),(5,6)\},&{}\{(0,7),(1,0),(4,5)\},\\ {} &{} \{(0,5),(1,0),(3,2)\},&{}\{(0,7),(4,0),(1,1)\},&{}\{(0,8),(4,0),(6,4)\},&{}\{(0,4),(2,0),(6,3)\},\\ {} &{} \{(0,0),(0,4),(6,4)\},&{}\{(0,0),(0,7),(4,7)\},&{}\{(0,0),(3,0),(3,8)\},&{}\{(0,0),(1,6),(1,7)\},\\ {} &{} \{(0,0),(5,0),(0,6)\},&{}\{(0,0),(0,2),(2,0)\},&{}\{(0,0),(1,0),(1,5)\},&{}\{(0,8),(3,0),(0,5)\},\\ {} &{} \{(0,4),(1,0),(4,1)\},&{}\{(0,0),(4,8),(6,1)\},&{}\{(0,0),(2,7),(6,3)\},&{}\{(0,6),(3,0),(6,3)\}. \\ \cdot &{} (n_1,n_2)=(19,17):\\ {} &{} \{(0,0),(1,1),(9,8)\},&{}\{(9,0),(8,1),(0,8)\},&{}\{(0,0),(9,1),(8,8)\},&{}\{(9,0),(0,1),(1,8)\},\\ {} &{} \{(0,0),(9,3),(7,8)\},&{}\{(9,0),(0,3),(2,8)\},&{}\{(0,0),(8,2),(5,8)\},&{}\{(8,0),(0,2),(3,8)\},\\ {} &{} \{(0,0),(9,5),(2,7)\},&{}\{(9,0),(0,5),(7,7)\},&{}\{(0,0),(4,7),(2,8)\},&{}\{(4,0),(0,7),(2,8)\},\\ {} &{} \{(0,0),(9,6),(6,7)\},&{}\{(9,0),(0,6),(3,7)\},&{}\{(0,0),(6,2),(3,7)\},&{}\{(6,0),(0,2),(3,7)\},\\ {} &{} \{(0,0),(5,5),(7,7)\},&{}\{(7,0),(2,5),(0,7)\},&{}\{(0,0),(9,4),(6,6)\},&{}\{(9,0),(0,4),(3,6)\},\\ {} &{} \{(0,0),(9,2),(5,7)\},&{}\{(9,0),(0,2),(4,7)\},&{}\{(0,0),(7,5),(6,8)\},&{}\{(7,0),(0,5),(1,8)\},\\ {} &{} \{(0,0),(5,2),(1,8)\},&{}\{(5,0),(0,2),(4,8)\},&{}\{(0,0),(8,1),(2,6)\},&{}\{(8,0),(0,1),(6,6)\},\\ {} &{} \{(0,0),(7,4),(8,6)\},&{}\{(8,0),(1,4),(0,6)\},&{}\{(4,0),(0,2),(3,6)\},&{}\{(0,0),(8,5),(1,6)\},\\ {} &{} \{(7,0),(0,3),(5,4)\},&{}\{(0,0),(8,3),(4,4)\},&{}\{(8,0),(2,1),(0,4)\},&{}\{(8,0),(0,3),(7,4)\},\\ {} &{} \{(0,0),(8,4),(3,8)\},&{}\{(0,0),(9,0),(0,7)\},&{}\{(0,0),(4,0),(4,8)\},&{}\{(0,0),(8,0),(0,5)\},\\ {} &{} \{(6,0),(3,0),(0,4)\},&{}\{(0,0),(2,4),(6,0)\},&{}\{(0,0),(5,3),(5,4)\},&{}\{(5,0),(0,3),(0,6)\},\\ {} &{} \{(0,0),(2,0),(4,3)\},&{}\{(0,0),(7,0),(3,3)\},&{}\{(0,0),(5,0),(6,4)\},&{}\{(0,0),(1,0),(7,3)\},\\ {} &{} \{(0,0),(0,2),(0,6)\},&{}\{(0,0),(1,5),(7,6)\},&{}\{(0,0),(4,1),(9,7)\},&{}\{(7,0),(0,6),(4,8)\},\\ {} &{} \{(4,0),(1,3),(0,8)\}. \end{array} \end{aligned}$$

Appendix: Base blocks in the proof of Lemma 6.14

$$\begin{aligned} \begin{array}{llllll} \cdot &{} (n_1,n_2)=(17,23): \\ {} &{} \{(0,0),(2,0),(5,0)\}, &{}\{(0,0),(1,0),(7,0)\}, &{}\{(0,0),(8,0),(4,2)\}, &{}\{(0,0),(4,0),(4,9)\},\\ {} &{} \{(0,0),(2,1),(4,11)\}, &{}\{(0,0),(0,1),(4,6)\}, &{}\{(4,0),(0,1),(0,9)\}, &{}\{(2,0),(4,4),(0,9)\},\\ {} &{} \{(4,0),(2,1),(0,11)\}, &{}\{(2,0),(4,3),(0,11)\}, &{}\{(0,0),(4,1),(2,8)\}, &{}\{(0,0),(0,2),(4,10)\},\\ {} &{} \{(0,0),(0,5),(2,11)\}, &{}\{(0,0),(4,4),(2,7)\}, &{}\{(0,0),(4,3),(2,9)\}, &{}\{(4,0),(0,4),(4,11)\},\\ {} &{} \{(0,0),(0,6),(0,10)\}, &{}\{(4,0),(2,4),(0,6)\}, &{}\{(2,0),(4,5),(0,8)\}, &{}\{(2,0),(0,5),(2,7)\},\\ {} &{} \{(4,0),(4,3),(0,10)\}.\\ \cdot &{} (n_1,n_2)=(41,17):\\ {} &{} \{(0,0),(3,0),(20,0)\}, &{}\{(0,0),(4,0),(16,0)\}, &{}\{(0,0),(1,0),(15,0)\}, &{}\{(0,0),(7,0),(18,0)\},\\ {} &{} \{(0,0),(5,0),(13,0)\}, &{}\{(0,0),(9,0),(19,0)\}, &{}\{(0,0),(6,0),(2,8)\}, &{}\{(0,0),(4,1),(4,8)\},\\ {} &{} \{(0,0),(0,1),(4,7)\}, &{}\{(0,0),(0,2),(4,4)\}, &{}\{(2,0),(4,0),(0,4)\}, &{}\{(2,0),(0,2),(4,5)\},\\ {} &{} \{(4,0),(0,3),(2,7)\}, &{}\{(2,0),(0,1),(4,6)\}, &{}\{(4,0),(0,6),(2,8)\}, &{}\{(0,0),(2,3),(0,8)\},\\ {} &{} \{(0,0),(0,6),(2,7)\}, &{}\{(4,0),(0,1),(0,5)\}, &{}\{(2,0),(2,3),(0,6)\}, &{}\{(4,0),(0,2),(0,7)\}.\\ \cdot &{} (n_1,n_2)=(41,23):\\ {} &{} \{(0,0),(7,0),(20,0)\}, &{}\{(0,0),(1,0),(19,0)\}, &{}\{(0,0),(14,0),(16,0)\}, &{}\{(0,0),(5,0),(15,0)\},\\ {} &{} \{(0,0),(9,0),(12,0)\}, &{}\{(0,0),(6,0),(17,0)\}, &{}\{(0,0),(8,0),(4,2)\}, &{}\{(0,0),(4,0),(4,9)\},\\ {} &{} \{(0,0),(2,1),(4,11)\}, &{}\{(0,0),(0,1),(4,6)\}, &{}\{(4,0),(0,1),(0,9)\}, &{}\{(2,0),(4,4),(0,9)\},\\ {} &{} \{(4,0),(2,1),(0,11)\}, &{}\{(2,0),(4,3),(0,11)\}, &{}\{(0,0),(4,1),(2,8)\}, &{}\{(0,0),(0,2),(4,10)\},\\ {} &{} \{(0,0),(0,5),(2,11)\}, &{}\{(0,0),(4,4),(2,7)\}, &{}\{(0,0),(4,3),(2,9)\}, &{}\{(4,0),(0,4),(4,11)\},\\ {} &{} \{(0,0),(0,6),(0,10)\}, &{}\{(4,0),(2,4),(0,6)\}, &{}\{(2,0),(4,5),(0,8)\}, &{}\{(2,0),(0,5),(2,7)\},\\ {} &{} \{(4,0),(4,3),(0,10)\}.\\ \cdot &{} (n_1,n_2)=(41,41):\\ {} &{} \{(0,0),(3,0),(20,0)\}, &{}\{(0,0),(4,0),(16,0)\}, &{}\{(0,0),(1,0),(15,0)\}, &{}\{(0,0),(7,0),(18,0)\},\\ {} &{} \{(0,0),(5,0),(13,0)\}, &{}\{(0,0),(9,0),(19,0)\}, &{}\{(0,0),(6,0),(2,20)\}, &{}\{(0,0),(4,1),(4,20)\},\\ {} &{} \{(0,0),(0,1),(4,19)\}, &{}\{(0,0),(0,2),(4,14)\}, &{}\{(0,0),(0,3),(4,16)\}, &{}\{(2,0),(4,0),(0,18)\},\\ {} &{} \{(2,0),(4,2),(0,19)\}, &{}\{(0,0),(4,3),(0,17)\}, &{}\{(0,0),(2,1),(4,15)\}, &{}\{(0,0),(2,3),(4,10)\},\\ {} &{} \{(0,0),(2,4),(2,17)\}, &{}\{(0,0),(4,2),(4,17)\}, &{}\{(0,0),(0,4),(4,11)\}, &{}\{(0,0),(0,5),(2,16)\},\\ {} &{} \{(0,0),(2,6),(2,18)\}, &{}\{(0,0),(4,4),(0,10)\}, &{}\{(2,0),(4,8),(0,12)\}, &{}\{(4,0),(2,1),(0,16)\},\\ {} &{} \{(0,0),(4,8),(0,16)\}, &{}\{(4,0),(0,3),(0,10)\}, &{}\{(2,0),(0,2),(0,13)\}, &{}\{(2,0),(0,3),(0,9)\},\\ {} &{} \{(2,0),(0,11),(2,20)\}, &{}\{(2,0),(4,10),(0,17)\}, &{}\{(2,0),(0,7),(4,13)\}, &{}\{(4,0),(0,5),(0,13)\},\\ {} &{} \{(2,0),(4,5),(0,20)\}, &{}\{(4,0),(0,1),(2,16)\}, &{}\{(4,0),(2,14),(0,19)\}, &{}\{(0,0),(4,5),(0,14)\},\\ {} &{} \{(4,0),(0,2),(0,11)\}, &{}\{(0,0),(4,9),(2,19)\}, &{}\{(0,0),(2,12),(0,18)\}, &{}\{(4,0),(2,4),(0,12)\}.\\ \cdot &{} (n_1,n_2)=(41,47):\\ {} &{} \{(0,0),(7,0),(20,0)\}, &{}\{(0,0),(1,0),(19,0)\}, &{}\{(0,0),(14,0),(16,0)\}, &{}\{(0,0),(5,0),(15,0)\},\\ {} &{} \{(0,0),(9,0),(12,0)\}, &{}\{(0,0),(6,0),(17,0)\}, &{}\{(0,0),(8,0),(4,2)\}, &{}\{(0,0),(4,1),(0,23)\},\\ {} &{} \{(0,0),(4,0),(4,21)\}, &{}\{(0,0),(0,2),(4,22)\}, &{}\{(0,0),(2,1),(4,23)\}, &{}\{(0,0),(0,1),(4,18)\},\\ {} &{} \{(0,0),(0,3),(4,13)\}, &{}\{(0,0),(0,4),(4,19)\}, &{}\{(0,0),(2,4),(4,9)\}, &{}\{(0,0),(4,5),(2,20)\},\\ {} &{} \{(0,0),(2,2),(4,14)\}, &{}\{(0,0),(0,5),(2,21)\}, &{}\{(0,0),(4,6),(2,18)\}, &{}\{(0,0),(2,3),(4,16)\},\\ {} &{} \{(0,0),(4,4),(0,20)\}, &{}\{(4,0),(0,3),(0,18)\}, &{}\{(0,0),(0,8),(4,11)\}, &{}\{(0,0),(2,6),(2,19)\},\\ {} &{} \{(0,0),(0,6),(2,23)\}, &{}\{(0,0),(4,12),(2,15)\}, &{}\{(0,0),(4,7),(0,22)\}, &{}\{(2,0),(4,7),(0,21)\},\\ {} &{} \{(2,0),(4,9),(0,16)\}, &{}\{(0,0),(4,8),(0,18)\}, &{}\{(0,0),(2,14),(0,16)\}, &{}\{(2,0),(4,11),(0,17)\},\\ {} &{} \{(2,0),(0,4),(0,23)\}, &{}\{(2,0),(0,6),(0,20)\}, &{}\{(4,0),(2,10),(0,21)\}, &{}\{(4,0),(0,9),(0,19)\},\\ {} &{} \{(4,0),(2,7),(0,20)\}, &{}\{(4,0),(0,5),(0,17)\}, &{}\{(2,0),(0,1),(0,8)\}, &{}\{(4,0),(4,11),(0,12)\},\\ {} &{} \{(2,0),(0,5),(0,22)\}, &{}\{(2,0),(4,8),(0,19)\}, &{}\{(2,0),(4,10),(0,18)\}, &{}\{(4,0),(0,4),(0,13)\},\\ {} &{} \{(4,0),(2,9),(0,23)\}. \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{llllll} \cdot &{} (47,17):\\ {} &{} \{(0,0),(1,0),(23,0)\}, &{}\{(0,0),(9,0),(21,0)\}, &{}\{(0,0),(3,0),(20,0)\}, &{}\{(0,0),(13,0),(15,0)\},\\ {} &{} \{(0,0),(5,0),(19,0)\}, &{}\{(0,0),(6,0),(16,0)\}, &{}\{(0,0),(7,0),(18,0)\}, &{}\{(0,0),(8,0),(4,2)\},\\ {} &{} \{(4,0),(2,1),(0,7)\}, &{}\{(0,0),(4,0),(0,8)\}, &{}\{(0,0),(0,1),(4,6)\}, &{}\{(2,0),(4,3),(0,8)\},\\ {} &{} \{(0,0),(4,1),(4,7)\}, &{}\{(2,0),(4,4),(0,7)\}, &{}\{(2,0),(2,3),(0,5)\}, &{}\{(4,0),(0,4),(4,7)\},\\ {} &{} \{(0,0),(4,4),(2,8)\}, &{}\{(0,0),(2,5),(2,7)\}, &{}\{(0,0),(2,1),(0,4)\}, &{}\{(0,0),(2,2),(4,8)\},\\ {} &{} \{(4,0),(0,1),(0,6)\}.\\ \cdot &{} (47,23):\\ {} &{} \{(0,0),(1,0),(17,0)\}, &{}\{(0,0),(2,0),(22,0)\}, &{}\{(0,0),(7,0),(18,0)\}, &{}\{(0,0),(15,0),(21,0)\},\\ {} &{} \{(0,0),(9,0),(12,0)\}, &{}\{(0,0),(10,0),(23,0)\}, &{}\{(0,0),(5,0),(19,0)\}, &{}\{(0,0),(8,0),(4,2)\},\\ {} &{} \{(0,0),(0,1),(4,11)\}, &{}\{(2,0),(4,1),(0,11)\}, &{}\{(0,0),(4,3),(0,11)\}, &{}\{(0,0),(4,1),(4,8)\},\\ {} &{} \{(0,0),(2,4),(4,6)\}, &{}\{(0,0),(0,3),(2,9)\}, &{}\{(2,0),(4,5),(0,9)\}, &{}\{(0,0),(4,0),(2,10)\},\\ {} &{} \{(0,0),(2,3),(0,9)\}, &{}\{(0,0),(0,4),(2,11)\}, &{}\{(4,0),(0,3),(0,11)\}, &{}\{(2,0),(0,3),(0,8)\},\\ {} &{} \{(0,0),(4,4),(2,8)\}, &{}\{(4,0),(2,7),(0,9)\}, &{}\{(4,0),(2,1),(0,6)\}, &{}\{(4,0),(4,6),(0,7)\},\\ {} &{} \{(0,0),(4,5),(0,10)\}, &{}\{(0,0),(0,2),(4,9)\}.\\ \cdot &{} (47,47):\\ {} &{} \{(0,0),(1,0),(17,0)\}, &{}\{(0,0),(2,0),(22,0)\}, &{}\{(0,0),(7,0),(18,0)\}, &{}\{(0,0),(15,0),(21,0)\},\\ {} &{} \{(0,0),(9,0),(12,0)\}, &{}\{(0,0),(10,0),(23,0)\}, &{}\{(0,0),(5,0),(19,0)\}, &{}\{(0,0),(8,0),(4,2)\},\\ {} &{} \{(0,0),(0,1),(4,23)\}, &{}\{(0,0),(2,1),(4,18)\}, &{}\{(0,0),(4,1),(4,20)\}, &{}\{(0,0),(0,2),(4,19)\},\\ {} &{} \{(0,0),(0,3),(4,16)\}, &{}\{(0,0),(4,3),(0,23)\}, &{}\{(0,0),(2,4),(2,21)\}, &{}\{(0,0),(0,4),(4,12)\},\\ {} &{} \{(2,0),(4,5),(0,23)\}, &{}\{(0,0),(4,5),(4,14)\}, &{}\{(0,0),(4,6),(2,23)\}, &{}\{(0,0),(4,0),(4,21)\},\\ {} &{} \{(0,0),(2,2),(4,15)\}, &{}\{(0,0),(2,3),(4,11)\}, &{}\{(0,0),(4,4),(2,18)\}, &{}\{(0,0),(0,6),(2,22)\},\\ {} &{} \{(0,0),(0,7),(2,19)\}, &{}\{(0,0),(0,8),(0,22)\}, &{}\{(0,0),(0,5),(2,20)\}, &{}\{(0,0),(4,10),(2,11)\},\\ {} &{} \{(0,0),(2,9),(0,20)\}, &{}\{(0,0),(4,9),(0,16)\}, &{}\{(4,0),(2,4),(0,17)\}, &{}\{(4,0),(0,3),(0,21)\},\\ {} &{} \{(4,0),(0,10),(2,20)\}, &{}\{(4,0),(0,6),(0,19)\}, &{}\{(4,0),(2,6),(0,22)\}, &{}\{(4,0),(2,7),(0,9)\},\\ {} &{} \{(4,0),(2,3),(0,13)\}, &{}\{(2,0),(4,6),(0,21)\}, &{}\{(2,0),(2,10),(0,22)\}, &{}\{(4,0),(2,5),(0,14)\},\\ {} &{} \{(2,0),(4,7),(0,18)\}, &{}\{(4,0),(2,8),(0,23)\}, &{}\{(4,0),(0,1),(0,12)\}, &{}\{(0,0),(4,7),(0,15)\},\\ {} &{} \{(4,0),(0,4),(0,16)\}, &{}\{(2,0),(4,14),(0,19)\}. \end{array} \end{aligned}$$

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Wang, L., Cai, L., Feng, T. et al. Geometric orthogonal codes and geometrical difference packings. Des. Codes Cryptogr. 90, 1857–1879 (2022). https://doi.org/10.1007/s10623-022-01078-4

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