Construction of orthogonal matrices using integer as elements for nonbinary codin

Chang, S.-H.

English

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T l i i s  paper presents three inctiiods of constmcting orthogonal signals 
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W l l 0 : c  mp l l tud?  levels are discreto,  but n o t  limited t o  binary: 
usinl: 1.1-sequence, (2)  method by inspection, md (3) recursive method. 
(1) m e t l i d  
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An effect ive s e t  of signals for u.r.2 in a channel with additive vhi te  G 2 x -  
sian noise i s  the orthogopal set .  I.!i?tho:is of constmctiog orthogonal con t imms  
wavc;omns are T; i dely studied. 
base3 primarily on Ha,dama,rd matrices. 
m t r ix  whose elements are the integers 4-1 cnd -1. 
orders have been c o n s t r ~ c t e d ~ ' ~ ~ ~  thrwch the generation of pseudo-randcm 
sequences of the types (1) maximum Icr,C;t!i sequences (m-sequences), (2) qud-  
r n t i c  residue sequence (or  I c g e n k r  scqucnce), (3) twin prime sequence, and 
(4.) E d 1  sequence. 
t i o n  of orthogonal matrices using intczcrs (o r  ra t ional  numbers) as elements, 
although t h e i r  uses i n  non-binary coding c m  be anticipated. E'urthermore, it 
i s  f e l t  t ha t  such study may bring the two areas of endeavor, discrete coding 
and waveform design, c loser  t o  each other. 
The construction of binary orthogonrj- codes i s  
A I!-Aailrtrd matrix i s  an orthogonal 
C 
Hadamard matrices of various 
It seems tha t  no such study has been made f o r  the construc- 
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'Jkcee methods are explored. They zrc s d z c d  as fo l l a ro .  
(I) N-Sequenccs Over GF(P), p = 3, 5, -.- 7 ,  11 
To i l l u s t  :ate t h i s  method by an cxmpl e, consider p = 5 and an irreducible 
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pririitive polyliomial of degree rn = 2 over GI?(>) 
W i t l i  t he  a id  0: the shif t  regis ter  c i rcu i t  sho~m i n  Fig. 1, it is  easy t o  see 
tha?, a typical sequence generated by the pol-ynodal i s  
a 2 W i t h  period r = 5 -1 = 24. By listins the Zbove sequence and 11 successive 
cycl.ic shifts of the sequence i n  12 Tars, md rctainiw only the first 12 
colims, then a 12 X 12 orthogond matrix using elements 0, f1,+2 i s  obtained. 
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1 0  2-1 2 2-2 0 1 2  1 1  
0 2-1 2 2-2 0 1 2  1 1 - 1  
2-1 2 2 - 2  0 1 2  1 1 - 1  0 
-1 2 2-2 0 1 2 1 1-1 0-2 
2 2-2 0 1 2  1 1 - 1  0-2 1 
2-2 0 1 2 1 1-1 0-2 1-2 
-2 0 1 2  1 1 - 1  0 - 2  1-2-2 
0 1 2 1 1-1 0 - 2  1-2-2 2 
1 2 1 1-1 0 - 2  1 - 2 - 2  2 0 
2 1 1 - 1  0 - 2  1-2-2 2 0-1 
1 1 -1 0 -2 1 -2 -2 2 0 -1 -2 
1-1 0 -2 1 - 2  -2 2 0 -1 -2 -1 
This method is eas i ly  exkeudcd t o  generate n X n o r t h o g o d  matrices, 
f i c l d  of W(p) oiito clcmcnto of intclt,crs, o r  rational numbers (Gee t ab le )  c m  
. 
2 0 1 7  0 0 0 
1 -1 1 1 3 
0 0 2 2 G 
2 
3 
2 / 3  1 1 3 
2 -1 4 -2/3 -2 
-2 -6 0 -  0 5 
1 1 6 -1 -3 
5 
2 -1 
-2 
-1 
a 
2 
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+< 
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be used t o  obtain n X n orthogonal nntr iccs  with 3, 7 and 11 e lemnts  with 1 :  
0 0 0 
1 1 2 
4 2 2 
3 3 6 a 4 4 
5 -112 -1 
6 112 
-4 7 
8 -3 
9 -2 -4 
11 
1 
-8 
-6 
10 -1 -2 
Table of Mapping Elements of GI?(?) t o  Intecers o r  Rationals f o r  the 
Construction of Orthogonal ; ht r ices  from t h e  M-&quencer, - 
The construction is based upon the propcrties' of the autocorrelation 
I function @(T) of the m-sequences of p elements (p = 3, 5, 7, 11) re lat ive t o  
cer; ain mapping q. 
m-sc pence,  namely, r = pm-L 
the  values of @(T) at T = 0 and T = r/2 differ i n  signs but equal in magnitude. 
Thei-efore, unlike the case f o r  p = 2, a s e w n t  equal t o  the half period of 
the  sequence i s  used for construction of the orthogonal matrices. 
f o r  cases p =X 7 and 11, @(T) assutws non-zero values under ordinary mapping 
for T omaller than a half period. 
z'cw ?.ppjng the eLemri.t;s of GF(7) a~il  GT(L1.) on to  Epecially chosen c e t s  of 
zat  1,oarzln or iiitcgoro. 
The autocorrelation function has the same period as the 
Under syrmr,ntrical mapping, as adopted here, 
Furthermore, 
Tlicsc RJY matored t o  zero by suitab1.e 
Bii21nz. psoccrlurcr: oppJ.icd t o  thc cases for  p > 11 
c 
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resu l t  i n  solutions in mapping of elemento of GF(L~), e tc  onto elzmento of 
i r ra t iona l  or complex field. 
(2) Construction by Inspection 
The follawing orthogonal matrices arc obtained by inspection. 
(1) 2 x 2 (3 level  o r  l e s s )  (2) 4 x 4 (7 level or l e s s )  
. (3) 8 x 8 (15 leve l  o r  less) 
A 
. 
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a b - c  d e f g h- 
-b a d c - f  e - h  Q 
c - d  a b - I :  h e - f  
-d -c -b a -11 -g f e 
-e f Q h (t -b c -d 
-f -e -h g b a -d -c 
-g h -e -f -c d a -b 
i-11  -g f -e d c b a - . 
By assigning suitable values t o  the l e t te rs ,  some of which may have the 
sam: value, orthogonal matrices of various elements can be constructed. 
(3)  Recursive Methods 
k t  A and B be two orthogonal matfices of s ize  n X n. 
recxrsive methods may be used t o  obtain new orthogonal matrices. 
Then the following 
(a) C = A.B size n X n, different elements from A or B. 
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(c )  C = [-; :]@A size 2n X 2n, different elementE aa A. 
A B  
size 2n X 2n, same elements as A and B 
combined prcwidcd AB = BAD 
('I = [-BT 03 . 
In  the last method, the matrix D is computed from: 
However, if AB = BA, then 
. , and 
rp 
. 
R e f €  rences 
1. 
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I1 N. Zierler, 
31-48, 1959, also i n  W. H. Kautz (editor), Linear Sequential Switching 
Circuits, Holden-Day, 1965. 
Lineax Recurring Sequences", J. SOc. Indust. Apple &lath., 1, 
2. E. F. Beckenbach (editor), Applied Combinatorial Mathematics, Chapter 13, 
Block Designs by M. H a l l ,  Jr., John Wiley and Sons, 1964. 
3 D  S. W. Golamb (editor), Digital Camnntnications with Space Applications, 
Chapter 4, Codes with Special Correlation by L. D. Baumert, Prentice-Hall, 
1964. 
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A note on non-binary orthogonal codes

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