Nearly orthogonal Latin squares are useful for conducting experiments eliminating heterogeneity in two directions and using different interventions each at each level.In this paper,some constructions of mutually nearl...Nearly orthogonal Latin squares are useful for conducting experiments eliminating heterogeneity in two directions and using different interventions each at each level.In this paper,some constructions of mutually nearly orthogonal Latin squares are provided.It is proved that there exist 3 MNOLS(2m) if and only if m ≥3 nd there exist 4 MNOLS(2m) if and only if m ≥4 with some possible exceptions.展开更多
A weakly pandiagonal Latin square of order n over the number set {0, 1, . . . , n-1} is a Latin square having the property that the sum of the n numbers in each of 2n diagonals is the same. In this paper, we shall pro...A weakly pandiagonal Latin square of order n over the number set {0, 1, . . . , n-1} is a Latin square having the property that the sum of the n numbers in each of 2n diagonals is the same. In this paper, we shall prove that a pair of orthogonal weakly pandiagonal Latin squares of order n exists if and only if n ≡ 0, 1, 3 (mod 4) and n≠3.展开更多
Denote by SFin(v) the set of all integer pairs (t, s) for which there exist three symmetric Latin squares of order v on the same set having fine structure (t, s). We completely determine the set SFin(2n) for a...Denote by SFin(v) the set of all integer pairs (t, s) for which there exist three symmetric Latin squares of order v on the same set having fine structure (t, s). We completely determine the set SFin(2n) for any integer n ≥ 5.展开更多
Two Latin squares of order v are r-orthogonal if their superposition produces exactly r distinct ordered pairs. The two squares are said to be r-orthogonal idempotent Latin squares and denoted by r-MOILS(v)if they a...Two Latin squares of order v are r-orthogonal if their superposition produces exactly r distinct ordered pairs. The two squares are said to be r-orthogonal idempotent Latin squares and denoted by r-MOILS(v)if they are all idempotent. In this paper, we show that for any integer v≥28, there exists an r-MOILS(v) if and only if r∈[v, v^2]/ {v + 1, v^2-1}.展开更多
In this paper, we give some constructions of self-conjugate self-orthogonal diagonal Latinsquares (SCSODLS). As an application of such constructions we disproof the conjecture aboutSCSODLS and show that there exist SC...In this paper, we give some constructions of self-conjugate self-orthogonal diagonal Latinsquares (SCSODLS). As an application of such constructions we disproof the conjecture aboutSCSODLS and show that there exist SCSODLS of order V, whenever w=1 (mod 12), with thepossible exception of v∈ {13, 85, 133}.展开更多
In recent years,a lot of XOR-based coding schemes have been developed to tolerate double disk failures in Redundant Array of Independent Disks (RAID) architectures,such as EVENODD-code,X-code,B-code and BG-HEDP. Despi...In recent years,a lot of XOR-based coding schemes have been developed to tolerate double disk failures in Redundant Array of Independent Disks (RAID) architectures,such as EVENODD-code,X-code,B-code and BG-HEDP. Despite those researches,the decades-old strategy of Reed-Solomon (RS) code remains the only popular space-optimal Maximum Distance Separable (MDS) code for all but the smallest storage systems. The reason is that all those XOR-based schemes are too difficult to be implemented,it mainly because the coding-circle of those codes vary with the number of disks. By contrast,the coding-circle of RS code is a constant. In order to solve this problem,we develop a new MDS code named Latin code and a cascading scheme based on Latin code. The cascading Latin scheme is a nearly MDS code (with only one or two more parity disks compared with the MDS ones). Nev-ertheless,it keeps the coding-circle of the basic Latin code (i.e. a constant) and the low encod-ing/decoding complexity similar to other parity array codes.展开更多
An n×n matrix A consisting of nonnegative integers is a general magic square of order n if thesum of elements in each row,column,and main diagonal is the same.A general magic square A of order n iscalled a magic ...An n×n matrix A consisting of nonnegative integers is a general magic square of order n if thesum of elements in each row,column,and main diagonal is the same.A general magic square A of order n iscalled a magic square,denoted by MS(n),if the entries of A are distinct.A magic square A of order n is normalif the entries of A are n^2 consecutive integers.Let A^*d denote the matrix obtained by raising each elementof A to the d-th power.The matrix A is a d-multimagic square,dcnoted by MS(n,d),if A^*e is an MS(n)for 1≤e≤d.In this paper we investigate the existence of normal bimagic squares of order 2u and prove that thereexists a normal bimagic square of order 2u,where u and 6 are coprime and u≥5.展开更多
Let N = {0, 1, ···, n-1}. A strongly idempotent self-orthogonal row Latin magic array of order n(SISORLMA(n) for short) based on N is an n × n array M satisfying the following properties:(1)...Let N = {0, 1, ···, n-1}. A strongly idempotent self-orthogonal row Latin magic array of order n(SISORLMA(n) for short) based on N is an n × n array M satisfying the following properties:(1) each row of M is a permutation of N, and at least one column is not a permutation of N;(2) the sums of the n numbers in every row and every column are the same;(3) M is orthogonal to its transpose;(4) the main diagonal and the back diagonal of M are 0, 1, ···, n-1 from left to right. In this paper, it is proved that an SISORLMA(n)exists if and only if n ? {2, 3}. As an application, it is proved that a nonelementary rational diagonally ordered magic square exists if and only if n ? {2, 3}, and a rational diagonally ordered magic square exists if and only if n ≠2.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
A Mendelsohn triple system of order v,MTS(v)for short,is a pair(X,B)where X is a v-set(of points)and B is a collection of cyclic triples on X such that every ordered pair of distinct points from X appears in exactly o...A Mendelsohn triple system of order v,MTS(v)for short,is a pair(X,B)where X is a v-set(of points)and B is a collection of cyclic triples on X such that every ordered pair of distinct points from X appears in exactly one cyclic triple of B.The cyclic triple(a,b,c)contains the ordered pairs(a,b),(b,c)and(c,a).An MTS(v)corresponds to an idempotent semisymmetric Latin square (quasigroup)of order v.An MTS(v)is called frame self-orthogonal,FSOMTS for short,if its associated semisymmetric Latin square is frame self-orthogonal.It is known that an FSOMTS(1~n)exists for all n≡1(mod 3)except n=10 and for all n≥15,n≡0(mod 3)with possible exception that n=18.In this paper,it is shown that(i)an FSOMTS(2~n)exists if and only if n≡0,1(mod 3)and n>5 with possible exceptions n ∈{9,27,33,39};(ii)an FSOMTS(3~n)exists if and only if n≥4,with possible exceptions that n ∈{6,14,18,19}.展开更多
基金Supported by the National Natural Science Foundations of China(Nos.11071207,11371308,11301457,11501181)
文摘Nearly orthogonal Latin squares are useful for conducting experiments eliminating heterogeneity in two directions and using different interventions each at each level.In this paper,some constructions of mutually nearly orthogonal Latin squares are provided.It is proved that there exist 3 MNOLS(2m) if and only if m ≥3 nd there exist 4 MNOLS(2m) if and only if m ≥4 with some possible exceptions.
基金Supported by National Natural Science Foundation of China(Grant Nos.61071221,10831002,11071207 and 11201407)Natural Science Foundation of Jiangsu Higher Education Institutions of China(Grant No.12KJD110007)Natural Science Foundation of Jiangsu Province(Grant No.BK2012245)
文摘A weakly pandiagonal Latin square of order n over the number set {0, 1, . . . , n-1} is a Latin square having the property that the sum of the n numbers in each of 2n diagonals is the same. In this paper, we shall prove that a pair of orthogonal weakly pandiagonal Latin squares of order n exists if and only if n ≡ 0, 1, 3 (mod 4) and n≠3.
基金Supported by National Natural Science Foundation of China (Grant No. 10771013)
文摘Denote by SFin(v) the set of all integer pairs (t, s) for which there exist three symmetric Latin squares of order v on the same set having fine structure (t, s). We completely determine the set SFin(2n) for any integer n ≥ 5.
基金Supported by the National Natural Science Foundation of China under Grant No.61373007,11371208Zhejiang Provincial Natural Science Foundation of China under Grant No.LY13F020039Sponsored by K.C.Wong Magna Fund in Ningbo University
文摘Two Latin squares of order v are r-orthogonal if their superposition produces exactly r distinct ordered pairs. The two squares are said to be r-orthogonal idempotent Latin squares and denoted by r-MOILS(v)if they are all idempotent. In this paper, we show that for any integer v≥28, there exists an r-MOILS(v) if and only if r∈[v, v^2]/ {v + 1, v^2-1}.
文摘In this paper, we give some constructions of self-conjugate self-orthogonal diagonal Latinsquares (SCSODLS). As an application of such constructions we disproof the conjecture aboutSCSODLS and show that there exist SCSODLS of order V, whenever w=1 (mod 12), with thepossible exception of v∈ {13, 85, 133}.
基金Supported in part by the National High Technology Re-search and Development Program of China (2008 AA01Z-401)the National Science Foundation of China (No.60903028)+1 种基金Doctoral Fund of Ministry of Education of China (20070055054)Science and Technology De-velopment Plan of Tianjin (08JCYBJC13000)
文摘In recent years,a lot of XOR-based coding schemes have been developed to tolerate double disk failures in Redundant Array of Independent Disks (RAID) architectures,such as EVENODD-code,X-code,B-code and BG-HEDP. Despite those researches,the decades-old strategy of Reed-Solomon (RS) code remains the only popular space-optimal Maximum Distance Separable (MDS) code for all but the smallest storage systems. The reason is that all those XOR-based schemes are too difficult to be implemented,it mainly because the coding-circle of those codes vary with the number of disks. By contrast,the coding-circle of RS code is a constant. In order to solve this problem,we develop a new MDS code named Latin code and a cascading scheme based on Latin code. The cascading Latin scheme is a nearly MDS code (with only one or two more parity disks compared with the MDS ones). Nev-ertheless,it keeps the coding-circle of the basic Latin code (i.e. a constant) and the low encod-ing/decoding complexity similar to other parity array codes.
基金This paper is supported by the National Natural Science Foundation of China(Nos.11871417,11501181)Science Foundation for Youths(Grant No.2014QK05)Ph.D.(Grant No.qd14140)of Henan Normal University.
文摘An n×n matrix A consisting of nonnegative integers is a general magic square of order n if thesum of elements in each row,column,and main diagonal is the same.A general magic square A of order n iscalled a magic square,denoted by MS(n),if the entries of A are distinct.A magic square A of order n is normalif the entries of A are n^2 consecutive integers.Let A^*d denote the matrix obtained by raising each elementof A to the d-th power.The matrix A is a d-multimagic square,dcnoted by MS(n,d),if A^*e is an MS(n)for 1≤e≤d.In this paper we investigate the existence of normal bimagic squares of order 2u and prove that thereexists a normal bimagic square of order 2u,where u and 6 are coprime and u≥5.
基金Supported by the National Natural Science Foundation of China(No.11271089)Guangxi Nature Science Foundation(No.2012GXNSFAA053001)+1 种基金Key Foundation of Guangxi Education Department(No.201202ZD012)Guangxi “Ba Gui” Team for Research and Innovation
文摘Let N = {0, 1, ···, n-1}. A strongly idempotent self-orthogonal row Latin magic array of order n(SISORLMA(n) for short) based on N is an n × n array M satisfying the following properties:(1) each row of M is a permutation of N, and at least one column is not a permutation of N;(2) the sums of the n numbers in every row and every column are the same;(3) M is orthogonal to its transpose;(4) the main diagonal and the back diagonal of M are 0, 1, ···, n-1 from left to right. In this paper, it is proved that an SISORLMA(n)exists if and only if n ? {2, 3}. As an application, it is proved that a nonelementary rational diagonally ordered magic square exists if and only if n ? {2, 3}, and a rational diagonally ordered magic square exists if and only if n ≠2.
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.
基金Research supported by NSFC 10371002Partially supported by National Science Foundation under Grant CCR-0098093
文摘A Mendelsohn triple system of order v,MTS(v)for short,is a pair(X,B)where X is a v-set(of points)and B is a collection of cyclic triples on X such that every ordered pair of distinct points from X appears in exactly one cyclic triple of B.The cyclic triple(a,b,c)contains the ordered pairs(a,b),(b,c)and(c,a).An MTS(v)corresponds to an idempotent semisymmetric Latin square (quasigroup)of order v.An MTS(v)is called frame self-orthogonal,FSOMTS for short,if its associated semisymmetric Latin square is frame self-orthogonal.It is known that an FSOMTS(1~n)exists for all n≡1(mod 3)except n=10 and for all n≥15,n≡0(mod 3)with possible exception that n=18.In this paper,it is shown that(i)an FSOMTS(2~n)exists if and only if n≡0,1(mod 3)and n>5 with possible exceptions n ∈{9,27,33,39};(ii)an FSOMTS(3~n)exists if and only if n≥4,with possible exceptions that n ∈{6,14,18,19}.