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New Constructions of Group-Invariant Butson Hadamard Matrices

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Tai Do Duc

Combinatorics · Submitted 23 Mar 2019

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this paper, we give three new constructions of $\text{BH}(G,h)$ matrices.

how it is stated · asserted 1.00 read the record →

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arXiv ID1903.09824 (version 4)
Pith IDOLQM5T6F, Pith's permanent record of this paper and its review
CitationTai Do Duc. "New Constructions of Group-Invariant Butson Hadamard Matrices." arXiv:1903.09824 (2019). https://pith.science/paper/OLQM5T6F
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,385 input tokens.

What the abstract itself states, as probabilities: claims first 0.82 · states limitation 0.14 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.64 · compares with prior results 0.78 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; .

Let $G$ be a finite group and let $h$ be a positive integer. A $\text{BH}(G,h)$ matrix is a $G$-invariant $|G|\times |G|$ matrix $H$ whose entries are complex $h$th roots of unity such that $HH^*=|G|I_{|G|}$, where $H^*$ denotes the complex conjugate transpose of $H$, and $I_{|G|}$ is the identity matrix of order $|G|$. In this paper, we give three new constructions of $\text{BH}(G,h)$ matrices. The first construction is the first known family of $\text{BH}(G,h)$ matrices in which $G$ does not need to be abelian. The second and the third constructions are two families of $\text{BH}(G,h)$ matrices in which $G$ is a finite local ring.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Let $G$ be a finite group and let $h$ be a positive integer. background 1.00 0.05 0.01 unclear 0.63
s1 A $\text{BH}(G,h)$ matrix is a $G$-invariant $|G|\times |G|$ matrix $H$ whose entries are complex $h$th roots of unity such that $HH^*=|G|I_{|G|}$, where $H^*$ denotes the complex conjugate transpose of $H$, and $I_{|G|}$ is the identity matrix of order $|G|$. background 1.00 0.05 0.03 asserted 0.98
s2 In this paper, we give three new constructions of $\text{BH}(G,h)$ matrices. contribution 1.00 0.97 0.77 asserted 1.00
s3 The first construction is the first known family of $\text{BH}(G,h)$ matrices in which $G$ does not need to be abelian. contribution 0.78 0.85 0.26 asserted 1.00
s4 The second and the third constructions are two families of $\text{BH}(G,h)$ matrices in which $G$ is a finite local ring. contribution 0.97 0.92 0.13 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.70 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Let $G$ be a finite group and let $h$ be a positive integer. A $\text{BH}(G,h)$ matrix is a $G$-invariant $|G|\times |G|$ matrix $H$ whose entries are complex $h$th roots of unity such that $HH^*=|G|I_{|G|}$, where $H^*$ denotes the complex conjugate transpose of $H$, and $I_{|G|}$ is the identity matrix of order $|G|$. In this paper, we give three new constructions of $\text{BH}(G,h)$ matrices. The first construction is the first known family of $\text{BH}(G,h)$ matrices in which $G$ does not need to be abelian. The second and the third constructions are two families of $\text{BH}(G,h)$ matrices in which $G$ is a finite local ring.

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