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Volume 38, Issue 2
On the Nonexistence of Partial Difference Sets by Projections to Finite Fields

Yue Zhou

Commun. Math. Res., 38 (2022), pp. 123-135.

Published online: 2022-02

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  • Abstract

In the study of (partial) difference sets and their generalizations in groups $G$, the most widely used method is to translate their definition into an equation over group ring $\mathbb{Z}[G]$ and to investigate this equation by applying complex representations of $G.$ In this paper, we investigate the existence of (partial) difference sets in a different way. We project the group ring equations in $\mathbb{Z}[G]$ to $\mathbb{Z}[N]$ where $N$ is a quotient group of $G$ isomorphic to the additive group of a finite field, and then use polynomials over this finite field to derive some existence conditions.

  • AMS Subject Headings

05B10, 05E30, 11T06

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COPYRIGHT: © Global Science Press

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@Article{CMR-38-123, author = {Zhou , Yue}, title = {On the Nonexistence of Partial Difference Sets by Projections to Finite Fields}, journal = {Communications in Mathematical Research }, year = {2022}, volume = {38}, number = {2}, pages = {123--135}, abstract = {

In the study of (partial) difference sets and their generalizations in groups $G$, the most widely used method is to translate their definition into an equation over group ring $\mathbb{Z}[G]$ and to investigate this equation by applying complex representations of $G.$ In this paper, we investigate the existence of (partial) difference sets in a different way. We project the group ring equations in $\mathbb{Z}[G]$ to $\mathbb{Z}[N]$ where $N$ is a quotient group of $G$ isomorphic to the additive group of a finite field, and then use polynomials over this finite field to derive some existence conditions.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2020-0049}, url = {http://global-sci.org/intro/article_detail/cmr/20267.html} }
TY - JOUR T1 - On the Nonexistence of Partial Difference Sets by Projections to Finite Fields AU - Zhou , Yue JO - Communications in Mathematical Research VL - 2 SP - 123 EP - 135 PY - 2022 DA - 2022/02 SN - 38 DO - http://doi.org/10.4208/cmr.2020-0049 UR - https://global-sci.org/intro/article_detail/cmr/20267.html KW - Partial difference set, strongly regular graph, finite field AB -

In the study of (partial) difference sets and their generalizations in groups $G$, the most widely used method is to translate their definition into an equation over group ring $\mathbb{Z}[G]$ and to investigate this equation by applying complex representations of $G.$ In this paper, we investigate the existence of (partial) difference sets in a different way. We project the group ring equations in $\mathbb{Z}[G]$ to $\mathbb{Z}[N]$ where $N$ is a quotient group of $G$ isomorphic to the additive group of a finite field, and then use polynomials over this finite field to derive some existence conditions.

Yue Zhou. (2022). On the Nonexistence of Partial Difference Sets by Projections to Finite Fields. Communications in Mathematical Research . 38 (2). 123-135. doi:10.4208/cmr.2020-0049
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