## Abstract

A p-ary function f in n variables is an l-form if f (tu) = t l f (u) for any nonzero t in Zp and u in Zn p. Let n be a positive even integer, p an odd prime, and l an element of {1, 2, . . . , p-1} provided that l ≠= p-1 if p > 3. Let f be a p-ary bent function in n variables of l -form with f (0) = 0 and gcd(l- 1, p- 1) = 1, and let Hl = {t l : T ϵ Z p}. We denote by G f,l the Cayley graph Cay(Zn p, ∪ϵs-Hl f-1(s)). Our main results are as follows: 1) if there is weakly regular p-ary bent f which is not regular, then l is 2; 2) if l = 2, then f is weakly regular p-ary bent if and only if the Cayley graph G f,l is strongly regular; 3) if l ≠= 2, then f is regular p-ary bent if and only if the Cayley graph G f,l is strongly regular; 4) G f,l can be replaced by Cay(Zn p, f-1(0)\{0}) in 2) and 3); and 5) amorphic association schemes are derived by using 2) and 3). We prove our main results by computing at most four distinct restricted eigenvalues of G f,l .

Original language | English |
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Article number | 8371636 |

Pages (from-to) | 676-684 |

Number of pages | 9 |

Journal | IEEE Transactions on Information Theory |

Volume | 65 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2019 |

### Bibliographical note

Funding Information:Manuscript received June 7, 2017; revised April 4, 2018; accepted May 19, 2018. Date of publication June 4, 2018; date of current version December 19, 2018. J. Y. Hyun was supported by the National Research Foundation of Korea Grant through the Korean Government (MEST) under Grant 2014R1A1A2A10054745. Y. Lee was supported in part by the Basic Science Research Program through the National Research Foundation of Korea (NRF) through the Ministry of Education under Grant 2009-0093827 and in part by the NRF Grant through the Korean Government (MEST) under Grant NRF-2017R1A2B2004574.

Publisher Copyright:

© 1963-2012 IEEE.

## Keywords

- (amorphic) association scheme
- p-ary bent function
- strongly regular graph