ORTHOGONAL DESIGNS

© Nickolay Balonin and Jennifer Seberry, 1.05.2014

Orthogonal designs catalogue and on-line algorithms

JENNIFER SEBERRY AND MIEKO YAMADA EXAMPLES

Definition. An orthogonal design of order n and type (s1, s2, ..., su) including positive integers, is a nxn matrix X with entries {0, ±x1, ..., ±xu} (the xi – commuting indeterminates) satisfying XTX=ωI, weight ω=Σi=1:usixi2. We write this as OD(n; s1, s2, ..., su) [1].

TWO BORDER TWO CIRCULANT SKEW CONFERENCE MATRIX CONSTRUCTION

EXAMPLES TO JENNIFER SEBERRY AND MIEKO YAMADA PAPER: n=3 (mod 4)

Example 1 For n=3, H has order 3×4=12

ORTHOGONAL DESIGN OD(4;1,3) AND PROPUS-MATRIX H12

Let us take attention, it is a Propus construction.

Example 2 For n=7, H has order 7×4=28

ORTHOGONAL DESIGN OD(4;1,3) AND PROPUS-MATRIX H28

Let us take attention, it is a Propus construction.

Example 3 For n=11, H has order 11×8=88

ORTHOGONAL DESIGN OD(8;1,7)

Example 4 For n=19, H has order 19×16=304

Example 5 For n=31, H has order 31×32=992

EXAMPLES TO JENNIFER SEBERRY AND MIEKO YAMADA PAPER: n=1 (mod 4)

Example 6 For n=5, H has order 5×16=80

LEGENDRE SYMBOLS: Q(5)

SUBMATRCES A, B, C, D

ORTHOGONAL DESIGN: OD(16;2,2,6,6)

Example 7 For n=13, H has order 13×128=1664

ORTHOGONAL DESIGN: OD(128;2,10,58,58)

Example 8 For n=17, H has order 17×64=1088

LEGENDRE SYMBOLS: Q(17)

SUBMATRCES A, B, C, D

ORTHOGONAL DESIGN: OD(64;2,2,30,30)

LEGENDRE SYMBOLS: Q(17) and OD(64;2,2,30,30)

Example 9 For n=17, H has order 17×64=1088

CHAIN OF ORTHOGONAL DESIGNS FROM OD(2;1,1) TO OD(64;2,2,30,30)

Example 10 For n=11, Balonin's algorithm to get Scarpis matrix H of order 11×12=132

HISTORIC EXAMPLE OF SKEW HADAMARD MATRIX H(44)

[1] Jennifer Seberry and Mieko Yamada, Hadamard matrices, Sequences, and Block Designs, An Existence Theorem for Hadamard Designs, Lemma 4.8, Cases 1, 2; P. 461.

[2] Gavin Cohen, David Rubie, Christos Koukouvinos, Stratis Kounias, Jennifer Seberry and Mieko Yamada, A survey of base sequences, disjoint complementary sequences and OD(4t; t, t, t, t), JCMCC, 5, (1989), 69–104.

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