TY - JOUR

T1 - Distribution of the exponents of primitive circulant matrices in the first four boxes of ℤn

AU - Bueno, Maria Isabel

AU - Fang, Kuan Ying

AU - Fuller, Samantha

AU - Furtado, Susana

N1 - Publisher Copyright:
© 2012, Mathematical Sciences Publishers.

PY - 2012

Y1 - 2012

N2 - We consider the problem of describing the possible exponents of n-by-n boolean primitive circulant matrices. It is well known that this set is a subset of [1, n −1] and not all integers in [1, n −1] are attainable exponents. In the literature, some attention has been paid to the gaps in the set of exponents. The first three gaps have been proven, that is, the integers in the intervals [n2+1, n−2], [n3+2, n2−2] and [n4+3, n −2] are not attainable exponents. Here we study the distribution 3 of exponents in between those gaps by giving the exact exponents attained there by primitive circulant matrices. We also study the distribution of exponents in between the third gap and our conjectured fourth gap. It is interesting to point out that the exponents attained in between the (i −1)-th and the i-th gap depend on the value of n mod i.

AB - We consider the problem of describing the possible exponents of n-by-n boolean primitive circulant matrices. It is well known that this set is a subset of [1, n −1] and not all integers in [1, n −1] are attainable exponents. In the literature, some attention has been paid to the gaps in the set of exponents. The first three gaps have been proven, that is, the integers in the intervals [n2+1, n−2], [n3+2, n2−2] and [n4+3, n −2] are not attainable exponents. Here we study the distribution 3 of exponents in between those gaps by giving the exact exponents attained there by primitive circulant matrices. We also study the distribution of exponents in between the third gap and our conjectured fourth gap. It is interesting to point out that the exponents attained in between the (i −1)-th and the i-th gap depend on the value of n mod i.

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U2 - 10.2140/involve.2012.5.187

DO - 10.2140/involve.2012.5.187

M3 - Article

AN - SCOPUS:85134677393

SN - 1944-4176

VL - 5

SP - 187

EP - 205

JO - Involve

JF - Involve

IS - 2

ER -