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The separation of binary relative three-weight codes and its applications

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Abstract

The separation and its applications of binary relative three-weight codes are studied. We first present a certain formula for computing the minimum values of the cardinality of separating coordinate position (CSCP) of some codeword sets for any binary linear code, and then with the above formula, we calculate all the minimum values of the CSCP of binary relative three-weight codes so as to determine their separation. By using the obtained separation of binary relative three-weight codes, some new binary separating codes are constructed. Based on a family of the minimum values of the CSCP of binary relative three-weight codes, we also determine all the minimal codewords and the trellis complexity of binary relative three-weight codes.

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Acknowledgements

The work of Y. Gao is supported in part by the National Key R&D Program of China under grant No. 2017YFB1400700.

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Authors and Affiliations

  1. School of Cyber Science and Technology, Beihang University, Beijing, 100191, People’s Republic of China

    Ying Gao

  2. Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Zihui Liu & Yiwei Liu

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  1. Ying Gao
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  2. Zihui Liu
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  3. Yiwei Liu
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Correspondence to Zihui Liu.

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Appendix: A proof of Theorem 4

Appendix: A proof of Theorem 4

We present in this Appendix the proof of Theorem 4.

Proof

It is necessary to get 𝜃2,1 and 𝜃2,2. The computing methods for 𝜃2,1 and 𝜃2,2 are similar. We only take 𝜃2,2 as an example, and our technique is to use Lemma 3 by computing all the possible |𝜃(T,T′)| (|T| = |T′| = 2). Noting that the codewords c2 and c3 in the formula given in Lemma 3 are symmetric, one may list all the possible distributions of independent codewords c1, c2 and c3 and the corresponding |𝜃(T,T′)| as follows.

  1. 1)

    c1, c2, c3 ∈ \(\mathcal {C}_{1}\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{1}}{4}\).

  2. 2)

    c1, c2 ∈ \(\mathcal {C}_{1}\) and \(c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{1}}{4}\).

  3. 3)

    c1, c2 ∈ \(\mathcal {C}_{1}\) and \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{1}}{4}\).

  4. 4)

    c2, c3 ∈ \(\mathcal {C}_{1}\) and \(c_{1}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{1}}{4}\).

  5. 5)

    c2, c3 ∈ \(\mathcal {C}_{1}\) and \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{1}}{4}\).

  6. 6)

    c1 ∈ \(\mathcal {C}_{1}\), \(c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), and \(c_{2}+c_{3}\in \mathcal {C}_{1}\). Then, \(|\theta (T,T^{\prime })|=\frac {4d_{2}-3d_{1}}{4}\).

  7. 7)

    c1 ∈ \(\mathcal {C}_{1}\), \(c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), and \(c_{2}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\). Then, \(|\theta (T,T^{\prime })|=\frac {2d_{2}-d_{1}}{4}\).

  8. 8)

    c1 ∈ \(\mathcal {C}_{1}\), \(c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\). Then, \(|\theta (T,T^{\prime })|=\frac {2d_{2}-d_{1}}{4}\).

  9. 9)

    c1 ∈ \(\mathcal {C}_{1}\), \(c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\) and \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\) and \(c_{2}+c_{3}\in \mathcal {C}_{1}\). Then, \(|\theta (T,T^{\prime })|=\frac {4d_{3}-3d_{1}}{4}\).

  10. 10)

    c1 ∈ \(\mathcal {C}_{1}\), \(c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\) and \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\) and \(c_{2}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\). Then, \(|\theta (T,T^{\prime })|=\frac {4d_{3}-2d_{2}-d_{1}}{4}\).

  11. 11)

    c1 ∈ \(\mathcal {C}_{1}\), \(c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\) and \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\) and \(c_{2}+c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\). Then, \(|\theta (T,T^{\prime })|=\frac {2d_{3}-d_{1}}{4}\).

  12. 12)

    c2 ∈ \(\mathcal {C}_{1}\), \(c_{1}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), and \(c_{1}+c_{3}\in \mathcal {C}_{1}\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{1}}{4}\).

  13. 13)

    c2 ∈ \(\mathcal {C}_{1}\), \(c_{1}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), and \(c_{1}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{1}}{4}\).

  14. 14)

    c2 ∈ \(\mathcal {C}_{1}\), \(c_{1}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{1}}{4}\).

  15. 15)

    c2 ∈ \(\mathcal {C}_{1}\), \(c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{1}}{4}\).

  16. 16)

    c2 ∈ \(\mathcal {C}_{1}\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), and \(c_{1}+c_{3}\in \mathcal {C}_{1}\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{1}}{4}\).

  17. 17)

    c2 ∈ \(\mathcal {C}_{1}\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), and \(c_{1}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{1}+d_{2}-d_{3}}{4}\).

  18. 18)

    c2 ∈ \(\mathcal {C}_{1}\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), and \(c_{1}+c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{1}}{4}\).

  19. 19)

    c1 ∈ \((\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), and \(c_{1}+c_{2}\in \mathcal {C}_{1}\) and \(c_{1}+c_{3}\in \mathcal {C}_{1}\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{1}}{4}\).

  20. 20)

    c1 ∈ \((\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), and \(c_{1}+c_{2}\in \mathcal {C}_{1}\) and \(c_{1}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{1}}{4}\).

  21. 21)

    c1 ∈ \((\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), and \(c_{1}+c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\) and \(c_{2}+c_{3}\in \mathcal {C}_{1}\). Then, \(|\theta (T,T^{\prime })|=\frac {2d_{2}-d_{1}}{4}\).

  22. 22)

    c1 ∈ \((\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), and \(c_{1}+c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\) and \(c_{2}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\) and \(c_{1}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\) and \(c_{1}+c_{2}+c_{3}\in \mathcal {C}_{1}\). Then, \(|\theta (T,T^{\prime })|=\frac {2d_{2}-d_{1}}{4}\).

  23. 23)

    c1 ∈ \((\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), and \(c_{1}+c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\) and \(c_{2}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\) and \(c_{1}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\) and \(c_{1}+c_{2}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{2}}{4}\).

  24. 24)

    c1 ∈ \((\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), and \(c_{1}+c_{2}\in \mathcal {C}_{1}\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{1}}{4}\).

  25. 25)

    c1 ∈ \((\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), and \(c_{1}+c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{2}}{4}\).

  26. 26)

    c2 ∈ \((\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), and \(c_{2}+c_{3}\in \mathcal {C}_{1}\). Then, \(|\theta (T,T^{\prime })|=\frac {2d_{2}-d_{1}}{4}\).

  27. 27)

    c2 ∈ \((\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), and \(c_{2}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{2}}{4}\).

  28. 28)

    \(c_{1}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), and \(c_{2}+c_{3}\in \mathcal {C}_{1}\). Then, \(|\theta (T,T^{\prime })|=\frac {4d_{3}-2d_{2}-d_{1}}{4}\).

  29. 29)

    \(c_{1}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{2}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), and \(c_{1}+c_{2}+c_{3}\in \mathcal {C}_{1}\). Then, \(|\theta (T,T^{\prime })|=\frac {4d_{3}-2d_{2}-d_{1}}{4}\).

  30. 30)

    \(c_{1}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{2}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), and \(c_{1}+c_{2}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\). Then, \(|\theta (T,T^{\prime })|=\frac {4d_{3}-3d_{2}}{4}\).

  31. 31)

    \(c_{1}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), and \(c_{2}+c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\). Then, \(|\theta (T,T^{\prime })|=\frac {2d_{3}-d_{2}}{4}\).

  32. 32)

    \(c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), and \(c_{1}+c_{3}\in \mathcal {C}_{1}\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{1}}{4}\).

  33. 33)

    \(c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{1}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), and \(c_{1}+c_{2}+c_{3}\in \mathcal {C}_{1}\). Then, \(|\theta (T,T^{\prime })|=\frac {2d_{2}-d_{1}}{4}\).

  34. 34)

    \(c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{1}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), and \(c_{1}+c_{2}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{2}}{4}\).

  35. 35)

    \(c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), and \(c_{1}+c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{2}}{4}\).

  36. 36)

    \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{1}+c_{2}\in \mathcal {C}_{1}\), and \(c_{1}+c_{3}\in \mathcal {C}_{1}\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{1}}{4}\).

  37. 37)

    \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{1}+c_{2}\in \mathcal {C}_{1}\), and \(c_{1}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{1}}{4}\).

  38. 38)

    \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{2}+c_{3}\in \mathcal {C}_{1}\), and \(c_{1}+c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\). Then, \(|\theta (T,T^{\prime })|=\frac {2d_{2}-d_{1}}{4}\).

  39. 39)

    \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{1}+c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), \(c_{1}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), and \(c_{2}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{2}}{4}\).

  40. 40)

    \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{1}+c_{2}\in \mathcal {C}_{1}\), and \(c_{1}+c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{1}}{4}\).

  41. 41)

    \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{1}+c_{2}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), and \(c_{1}+c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{2}}{4}\).

  42. 42)

    \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{2}+c_{3}\in \mathcal {C}_{1}\), and \(c_{1}+c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\). Then, \(|\theta (T,T^{\prime })|=\frac {2d_{3}-d_{1}}{4}\).

  43. 43)

    \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{2}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\), and \(c_{1}+c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\). Then, \(|\theta (T,T^{\prime })|=\frac {2d_{3}-d_{2}}{ 4}\).

  44. 44)

    \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{1}+c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{1}+c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{2}+c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), and \(c_{1}+c_{2}+c_{3}\in \mathcal {C}_{1}\). Then, \(|\theta (T,T^{\prime })|=\frac {2d_{3}-d_{1}}{4}\).

  45. 45)

    \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{1}+c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{1}+c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{2}+c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), and \(c_{1}+c_{2}+c_{3}\in (\mathcal {C}_{2}\backslash \mathcal {C}_{1})\). Then, \(|\theta (T,T^{\prime })|=\frac {2d_{3}-d_{2}}{4}\).

  46. 46)

    \(c_{1}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{1}+c_{2}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{1}+c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), \(c_{2}+c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\), and \(c_{1}+c_{2}+c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\). Then, \(|\theta (T,T^{\prime })|=\frac {d_{3}}{4}\).

To check each of the above cases, we take case 3) as an example. Since c1, c2 ∈ \(\mathcal {C}_{1}\) and \(c_{3}\in (\mathcal {C}\backslash \mathcal {C}_{2})\) and \(\mathcal {C}_{1}\subset \mathcal {C}_{2}\) are subcodes of \(\mathcal {C}\), one may get \(c_{1}+c_{2}\in \mathcal {C}_{1}\), c1 + c3 ∈ \((\mathcal {C}\backslash \mathcal {C}_{2})\), c2 + c3 ∈ \((\mathcal {C}\backslash \mathcal {C}_{2})\), and c1 + c2 + c3 ∈ \((\mathcal {C}\backslash \mathcal {C}_{2})\). Thus, w(c1) = w(c2) = d1, w(c3) = d3, w(c1 + c2) = d1, w(c1 + c3) = d3, w(c2 + c3) = d3, and w(c1 + c2 + c3) = d3. Then, \(|\theta (T,T^{\prime })|=\frac {d_{1}}{4}\) by using the formula in Lemma 3.

Making use of the above cases, one gets that

$$\begin{array}{@{}rcl@{}} \theta_{2,2}&=&\min_{T,T^{\prime}}|\theta(T,T^{\prime})|\\ &=&\min\left\{\frac{d_{1}}{4},\frac{4d_{2}-3d_{1}}{4}, \frac{2d_{2}-d_{1}}{4}, \frac{4d_{3}-3d_{1}}{4}, \frac{4d_{3}-2d_{2}-d_{1}}{4}, \frac{2d_{3}-d_{1}}{4}, \right.\\ &&\quad\quad \left. \frac{d_{1}+d_{2}-d_{3}}{4}, \frac{d_{2}}{4}, \frac{4d_{3}-3d_{2}}{4}, \frac{2d_{3}-d_{2}}{4}, \frac{d_{3}}{4}\right\}. \end{array} $$
(12)

The theorem follows by (9). □

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Gao, Y., Liu, Z. & Liu, Y. The separation of binary relative three-weight codes and its applications. Cryptogr. Commun. 11, 979–992 (2019). https://doi.org/10.1007/s12095-018-0339-9

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