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
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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 -