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

Yue Zhou

DOI: 10.4208/cmr.2020-0049

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.

  • Keywords

Partial difference set, strongly regular graph, finite field

  • AMS Subject Headings

05B10, 05E30, 11T06

  • Copyright

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.

Zhou , Yue. (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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