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Isomorphism of maximum length circuit codes

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Abstract

Douglas (J Comb Theory 6(4):323–339, 1969) proved that for odd spread k and dimension \(d={\frac{1}{2}(3k+3)}\) all maximum length (d, k) circuit codes are isomorphic. Here we establish a similar result, showing that for spread k even \(\ge 4\) and dimension \(d=\frac{1}{2}(3k+4)\) all maximum length symmetric circuit codes are isomorphic, and provide an explicit construction for such codes. We also extend a recent result of Byrnes (Des Codes Cryptogr 87(11):2671–2681, 2019) to give an exact formula for the maximum length of a symmetric circuit code with a long bit run. Numerous examples investigate the conditions under which all maximum length circuit codes are isomorphic, showing that for k even \(\ge 4\) and \(d=\frac{1}{2}(3k+4)\) symmetry is a necessary condition, and that even with symmetry, isomorphism is not guaranteed when k is odd \(\ge 9\) and \(d=\frac{1}{2}(3k+5)\).

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Notes

  1. Extending this proof beyond the symmetric case requires also demonstrating asymmetric \((\frac{1}{2}(3k+3),k)\) circuit codes of length \(4k+4\) do not exist, which we have not done.

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Acknowledgements

The author wishes to thank two anonymous referees for their thoughtful criticism and detailed feedback, which significantly improved the final version of this article. In particular, feedback from both referees led to the the investigation of Lemma 4.1, which greatly strengthened Theorem 1.4 compared to the original manuscript.

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, Maryland, 21218, USA

    Kevin M. Byrnes

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  1. Kevin M. Byrnes
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Correspondence to Kevin M. Byrnes.

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Communicated by J. H. Koolen.

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Appendix: Examples of circuit codes

Appendix: Examples of circuit codes

Throughout this article, we consider (d, k) circuit codes of the form: k odd and \(d=\frac{1}{2}(3k+3)\); k even \(\ge 4\) and \(d=\frac{1}{2}(3k+4)\); k odd \(\ge 9\) and \(d=\frac{1}{2}(3k+5)\). Table 1 presents all of the valid (d, k) pairs of these forms for small values of k, where an “X” indicates an invalid (d, k) combination.

Table 1 Valid (d, k) pairs for small values of k
Full size table

Below, we list the transition sequence of a maximum length symmetric (d, k) circuit code for each (d, k) pair in Table 1. The circuit codes are listed in the form (d, k, N) where N denotes the length of the circuit code (equal to K(d, k) for each of the examples presented).

(3, 1, 8)

1, 2, 3, 2, 1, 2, 3, 2

(6, 3, 16)

1, 2, 3, 4, 5, 2, 6, 4, 1, 2, 3, 4, 5, 2, 6, 4

(8, 4, 22)

1, 2, 3, 4, 5, 6, 7, 2, 4, 8, 6, 1, 2, 3, 4, 5, 6, 7, 2, 4, 8, 6

(9, 5, 24)

1, 2, 3, 4, 5, 6, 7, 2, 8, 4, 9, 6, 1, 2, 3, 4, 5, 6, 7, 2, 8, 4, 9, 6

(11, 6, 30)

1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 4, 10, 6,11, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 4, 10, 6, 11, 8

(12, 7, 32)

1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 10, 4, 11, 6, 12, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 10, 4, 11, 6, 12, 8

(14, 8, 38)

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 2, 4, 12, 6, 13, 8, 14, 10,

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 2, 4, 12 ,6, 13, 8, 14, 10

(15, 9, 40)

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 2, 12, 4, 13, 6, 14, 8, 15, 10,

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 2, 12, 4, 13, 6, 14, 8, 15, 10

(16, 9, 44)

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 2, 4, 6, 14, 8, 15, 10, 16, 12,

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 2, 4, 6, 14, 8, 15, 10, 16, 12

(17, 10, 46)

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 2, 4, 14, 6, 15, 8, 16, 10, 17, 12,

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 2, 4, 14, 6, 15, 8, 16, 10, 17, 12

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Byrnes, K.M. Isomorphism of maximum length circuit codes. Des. Codes Cryptogr. 90, 835–850 (2022). https://doi.org/10.1007/s10623-021-01005-z

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