TY - JOUR

T1 - Upper bounds on cyclotomic numbers

AU - Betsumiya, Koichi

AU - Hirasaka, Mitsugu

AU - Komatsu, Takao

AU - Munemasa, Akihiro

N1 - Funding Information:
Corresponding author. E-mail addresses: betsumi@cc.hirosaki-u.ac.jp (K. Betsumiya), hirasaka@pusan.ac.kr (M. Hirasaka), komatsu@cc.hirosaki-u.ac.jp (T. Komatsu), munemasa@math.is.tohoku.ac.jp (A. Munemasa). 1 The second author thanks the support from the grant represented by the third author when the second author stayed at Hirosaki University from April 22–27 in 2011. 2 The third author is supported in part by the Grant-in-Aid for Scientific Research (C) (No. 22540005), the Japan Society for the Promotion of Science.

PY - 2013/1/1

Y1 - 2013/1/1

N2 - In this article, we give upper bounds for cyclotomic numbers of order e over a finite field with q elements, where e is a positive divisor of q - 1. In particular, we show that under certain assumptions, cyclotomic numbers are at most ⌈k2⌉, and the cyclotomic number (0, 0) is at most ⌈k2⌉-1, where k=(q-1)/e. These results are obtained by using a known formula for the determinant of a matrix whose entries are binomial coefficients.

AB - In this article, we give upper bounds for cyclotomic numbers of order e over a finite field with q elements, where e is a positive divisor of q - 1. In particular, we show that under certain assumptions, cyclotomic numbers are at most ⌈k2⌉, and the cyclotomic number (0, 0) is at most ⌈k2⌉-1, where k=(q-1)/e. These results are obtained by using a known formula for the determinant of a matrix whose entries are binomial coefficients.

KW - Cyclotomic numbers

UR - http://www.scopus.com/inward/record.url?scp=84869090818&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84869090818&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2012.06.045

DO - 10.1016/j.laa.2012.06.045

M3 - Article

AN - SCOPUS:84869090818

SN - 0024-3795

VL - 438

SP - 111

EP - 120

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 1

ER -