bibkey: chirvasitucusick2024quadraticrotation
authors: A. Chirvasitu; T. W. Cusick
year: 2024
title: 'Quadratic rotation symmetric Boolean functions'
doi: 10.1016/j.dam.2023.10.010
url: https://arxiv.org/abs/2304.12734v1
claim: Section 6 defines the binomial rules matrices R(i,j) and Conjecture 6.1 proposes their scaled-root-of-unity spectrum and order 2 lcm(4, i-j, i+j).
strata_touched:
  - D5/S3/Combinatorics/BooleanFunctions/ChirvasituCusickRulesMatrixOrder
license: citation-only
triage: anchor

Quadratic rotation symmetric Boolean functions

A. Chirvasitu and T. W. Cusick, Quadratic rotation symmetric Boolean functions, Discrete Applied Mathematics 343 (2024), 91–105.

Verified locator

DOI: https://doi.org/10.1016/j.dam.2023.10.010

Source: https://arxiv.org/abs/2304.12734v1

Source statement

Conjecture 6.1, page 19:

For i > j ≥ 1 the eigenvalues of the matrix R(i, j) of (6-3) are all √2-scaled roots of unity. Moreover, the multiplicative order of R(i,j)/√2 is precisely 2 · lcm(4, i − j, i + j).

Section 6, page 18, defines $H_0=\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$ and $H_1=\left(\begin{array}{cc}1& 1\\ -1& 1\end{array}\right)$. On page 19:

by πₖ(M) (or πₖM) the matrix obtained by appending k all-0 columns after each column of M (‘π’ is meant to stand for ‘pad’);

and by ρₖ(M) (or ρₖM) the matrix obtained by rotation all rows of M simultaneously rightward k times.

Specifically, the rows in (6-2) alternate in whether or not an H₁ is substituted in batches of 2ʲ⁻¹:

Equation (6-3) starts with $H_0$ and stacks the successive rotated padded blocks. The source states: “Again, this is a 2ⁱ × 2ⁱ matrix.” Zero-based block and row indices correspond by $(k,c)\mapsto c+2k$.

The formal refutation concerns the order conjunct at $(8,4)$. It does not establish the exact order 48 or decide the universal eigenvalue conjunct. Completeness of the search for a prior published settlement is ASSUMED-UNVERIFIED.

Experiment entry

The independent checker is in trureturing-experiments, commit f6f05757e065d45c0f14bce4085f46b713d78896, SHA-256 f364b6795d229f1352f9ff59f297ef56fe0d08f4b7dbf2ef2ae3933f85e57e78. Running OPENBLAS_NUM_THREADS=1 VECLIB_MAXIMUM_THREADS=1 python3 docs/reports/chirvasitu-cusick-2023-rules-matrix-order/check.py in that repository exits 0 with RESULT PASS. The matrix powers at $(8,4)$ use integer arithmetic; the broader order scan uses floating arithmetic.