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Group invariant weighing matrices

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Abstract

We investigate the existence problem of group invariant matrices using algebraic approaches. We extend the usual concept of multipliers to group rings with cyclotomic integers as coefficients. This concept is combined with the field descent method and rational idempotents to develop new non-existence results.

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Acknowledgements

I am grateful to my advisor, Bernhard Schmidt for his helpful comments and discussions. Furthermore, I would like to thank the anonymous referees for their useful suggestions. I would also like to thank Artacho et al. [11, Remark 4.4 and 4.5] for pointing out the errors in the updated Strassler’s table (Table 3) in an earlier version of this paper.

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  1. School of Computer Science and Engineering, Nanyang Technological University, Singapore, Singapore

    Ming Ming Tan

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  1. Ming Ming Tan
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Correspondence to Ming Ming Tan.

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Communicated by K. T. Arasu.

Appendix: Updated Strassler’s table

Appendix: Updated Strassler’s table

Here, we provide the most updated version of Strassler’s table, see Table 3. We incorporated all latest results known so far, see [5, 6, 10, 26, 27, 35, 42]. Previously open cases that we have settled in this paper are labeled with “N”. Those are CW(v, n) for the following pairs of (v, n): (60, 36),  (120, 36),  (138, 36),  (155, 36),  (184, 36),  (128, 49),  (112, 64),  (147, 64),  (184, 64),  (105, 81),  (117, 81),  (184, 81),  (133, 100),  (154, 100),  (158, 100),  (160, 100),  (176, 100),  (190, 100),  and (192, 100). In the table, we also indicate which cases of CW(v, n) with \(n > 25\) that can be proved to be non-existent by the results presented in this paper. All CW(v, n)s with \(n \le 25\) have been completely classified, see [1, 8, 9, 18, 19, 38]. We use the following labels: (A) for Theorem 4.2, (B) for Theorem 4.6, (C) for Theorem 4.10, (D) for Theorem 4.14, (E) for Theorem 4.16, (F) for Theorem 4.21, (G) for Theorem 4.3 (H) for Theorem 4.23, (I) for Theorem 4.24, (J) for Theorem 4.25.

Table 3 Updated Strassler’s table of circulant weighing matrices
Full size table

In addition, we solved some open cases in the table of group invariant weighing matrices of [3], where the group is abelian but non-cyclic. Table 4 summarizes our results. Note that an abelian group \(G = C_{v_1} \times C_{v_2} \times \ldots \times C_{v_r}\) is represented by \([v_1, v_2, \ldots , v_r]\) in the table.

Table 4 Non-existence of G-invariant weighing matrices W(|G|, n)
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Tan, M.M. Group invariant weighing matrices. Des. Codes Cryptogr. 86, 2677–2702 (2018). https://doi.org/10.1007/s10623-018-0466-5

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