Let Fd be the finite field with d elements.Extraordinary subgroups in Fd×Fd play an important role in the field of quantum information theory,especially for the study of mutually unbiased bases.Recently,Ghiu et a...Let Fd be the finite field with d elements.Extraordinary subgroups in Fd×Fd play an important role in the field of quantum information theory,especially for the study of mutually unbiased bases.Recently,Ghiu et al.introduced the concept of supersquare of order d which is related to extraordinary subgroups.They have given a method of construction of the mutually orthogonal supersquares,and determined all the complete sets of mutually orthogonal extraordinary supersquares of order 4.In this article,we present the construction of a complete set of mutually orthogonal extraordinary supersquares of order pn where p is a prime.We also determine all the complete sets of mutually orthogonal extraordinary supersquares of order 9.展开更多
A Latin squares of order v with ni missing sub-Latin squares （holes） of order hi （1 〈= i 〈 k）, which are disjoint and spanning （i.e. ∑k i=l1 nihi = v）, is called a partitioned incomplete Latin squares and de...A Latin squares of order v with ni missing sub-Latin squares （holes） of order hi （1 〈= i 〈 k）, which are disjoint and spanning （i.e. ∑k i=l1 nihi = v）, is called a partitioned incomplete Latin squares and denoted by PILS. The type of PILS is defined by （h1n1 h2n2…hknk ）. If any two PILS inaset of t PILS of type T are orthogonal, then we denote the set by t-HMOLS（T）. It has been proved that 3-HMOLS（2n31） exist for n ≥6 with 11 possible exceptions. In this paper, we investigate the existence of 3-HMOLS（2nu1） with u ≥ 4, and prove that 3-HMOLS（2~u1） exist if n ≥ 54 and n ≥7/4u ＋ 7.展开更多
基金This paper is supported by the National Natural Science Foundation of China(Nos.12071226,11931006,12101414).
文摘Let Fd be the finite field with d elements.Extraordinary subgroups in Fd×Fd play an important role in the field of quantum information theory,especially for the study of mutually unbiased bases.Recently,Ghiu et al.introduced the concept of supersquare of order d which is related to extraordinary subgroups.They have given a method of construction of the mutually orthogonal supersquares,and determined all the complete sets of mutually orthogonal extraordinary supersquares of order 4.In this article,we present the construction of a complete set of mutually orthogonal extraordinary supersquares of order pn where p is a prime.We also determine all the complete sets of mutually orthogonal extraordinary supersquares of order 9.
基金Research supported by National Natural Science Foundation of China under Grant No. 60873267Zhejiang Provincial Natural Science Foundation of China under Grant No. Y607026sponsored by K. C. Wong Magna Fund at Ningbo University
文摘A Latin squares of order v with ni missing sub-Latin squares （holes） of order hi （1 〈= i 〈 k）, which are disjoint and spanning （i.e. ∑k i=l1 nihi = v）, is called a partitioned incomplete Latin squares and denoted by PILS. The type of PILS is defined by （h1n1 h2n2…hknk ）. If any two PILS inaset of t PILS of type T are orthogonal, then we denote the set by t-HMOLS（T）. It has been proved that 3-HMOLS（2n31） exist for n ≥6 with 11 possible exceptions. In this paper, we investigate the existence of 3-HMOLS（2nu1） with u ≥ 4, and prove that 3-HMOLS（2~u1） exist if n ≥ 54 and n ≥7/4u ＋ 7.
