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Let K be a set of positive integers and let G be an additive group. A ( G , K , 1 ) difference packing is a set of subsets of G with sizes from K whose list of differences covers every element of G at most once. It is balanced if the number of blocks of size k ∈ K does not depend on k. In this paper, we determine a balanced ( Z 4 u × Z 8 v , { 4 , 5 } , 1 ) difference packing of the largest possible size whenever uv is odd. The corresponding optimal balanced ( 4 u , 8 v , { 4 , 5 } , 1 ) optical orthogonal signature pattern codes are also obtained.

The concept of a ( q , k , λ , t ) almost difference family (ADF for short) was introduced by Ding and Yin as a useful generalization of the concept of an almost difference set. Some results had been obtained for the existences of ( q , K , λ ) -ADFs, where K = { k 1 , k 2 , ź , k r } is a set of positive integers. In this note, new cyclic ( q , K , λ ) -ADFs are obtained via cyclotomy.

Let W = { w 1 , w 2 , ? , w r } be an ordering of a set of r integers greater than 1, ? a = ( λ a ( 1 ) , λ a ( 2 ) , ? , λ a ( r ) ) be an r -tuple of positive integers, λ c be a positive integer, and Q = ( q 1 , q 2 , ? , q r ) be an r -tuple of positive rational numbers whose sum is 1. In 1996, Yang introduced variable-weight optical orthogonal code ( ( n , W , ? a , λ c , Q ) -OOC) for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. Some work had been done on the constructions of optimal ( n , { 3 , 4 } , ? a , 1 , Q ) -OOCs with unequal auto- and cross-correlation constraints. In this paper, we focus our main attention on ( n , { 3 , 5 } , ? a , 1 , Q ) -OOCs, where ? a ? { ( 1 , 2 ) , ( 2 , 1 ) , ( 2 , 2 ) } . Tight upper bounds on the maximum code size of an ( n , { 3 , 5 } , ? a , 1 , Q ) -OOC are obtained, and infinite classes of optimal balanced ( n , { 3 , 5 } , ? a , 1 ) -OOCs are constructed.

Let W = { w 1 , ? , w r } be an ordering of a set of r integers greater than 1, ? a = ( λ a ( 1 ) , ? , λ a ( r ) ) be an r -tuple of positive integers, λ c be a positive integer, and Q = ( q 1 , ? , q r ) be an r -tuple of positive rational numbers whose sum is 1. In 1996, Yang introduced variable-weight optical orthogonal code ( ( n , W , ? a , λ c , Q ) -OOC) for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. Most existing works on variable-weight optical orthogonal codes assume that λ a ( 1 ) , ? , λ a ( r ) = λ c = 1 . In this paper, new balanced ( n , { 3 , 4 } , ? a , 1 ) -OOCs are constructed, where ? a ? { ( 1 , 2 ) , ( 2 , 1 ) , ( 2 , 2 ) } .

In 1996, Yang introduced variable-weight optical orthogonal code for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. Let W = { w 1 , … , w r } be an ordering of a set of r integers greater than 1 , λ be a positive integer ( auto- and cross-correlation parameter ), and Q = ( q 1 , … , q r ) be an r -tuple ( weight distribution sequence ) of positive rational numbers whose sum is 1 . A ( v , W , λ , Q ) variable-weight optical orthogonal code ( ( v , W , λ , Q ) -OOC) is a collection of ( 0 , 1 ) sequences with weights in W , auto- and cross-correlation parameter λ . Some work has been done on the construction of optimal ( v , W , 1 , Q ) -OOCs, while little is known on the construction of ( v , W , λ , Q ) -OOCs with λ ≥ 2 . It is well known that ( v , W , λ , Q ) -OOCs with λ ≥ 2 have much bigger cardinality than those of ( v , W , 1 , Q ) -OOCs for the same v , W , Q . In this paper, a new upper bound on the number of codewords of ( v , W , λ , Q ) -OOCs is given, and infinite classes of optimal ( v , { 3 , 4 } , 2 , Q ) -OOCs are constructed.

In this note, a doubly magic rectangle is introduced to construct a doubly pandiagonal magic square. A product construction for doubly magic rectangles is also presented. Infinite classes of doubly pandiagonal magic squares are then obtained, and an answer to problem 22 of [G. Abe, Unsolved problems on magic squares, Discrete Math. 127 (1994) 3] is given.

Generalized Steiner systems GS(2,k,v,g) were first introduced by Etzion and used to construct optimal constant weight codes over an alphabet of size g+1 with minimum Hamming distance 2k−3, in which each codeword has length v and weight k. As to the existence of a GS(2,k,v,g), a lot of work has been done for k=3, while not so much is known for k=4. The notion k-∗GDD was first introduced by Chen et al. and used to construct GS(2,3,v,6). The necessary condition for the existence of a 4-∗GDD(6v) is v≥14. In this paper, it is proved that there exists a 4-∗GDD(6v) for any prime power v≡3,5,7(mod8) and v≥19. By using this result, the known results on the existence of optimal quaternary constant weight codes are then extended.

An (n,M,d;q) code is called equidistant code if the Hamming distance between any two codewords is d. It was proved that for any equidistant (n,M,d;q) code, d⩽nM(q-1)/(M-1)q(=dopt, say). A necessary condition for the existence of an optimal equidistant code is that dopt be an integer. If dopt is not an integer, i.e. the equidistant code is not optimal, then the code with d=⌊dopt⌋ is called good equidistant code, which is obviously the best possible one among equidistant codes with parameters n,M and q. In this paper, some constructions of good equidistant codes from balanced arrays and nested BIB designs are described.

Let X^1 be the m-vector (-r,-r+1,...,-1,0,1,...,r-1,r), m=2r+1, and X^2,...,X^n be permutations of X^1. Then X^1,X^2,...,X^n is said to be an additive sequence of permutations (ASP) of order m and length n if the vector sum of every subsequence of consecutive permutations is again a permutation of X^1. ASPs had been extensively studied and used to construct perfect difference families. In this paper, ASPs are used to construct perfect difference families and properly centered permutation matrices (which are related to radar arrays). More existence results on perfect difference families and properly centered permutation matrices are obtained.