milewski sequences revisited, and its generalization
TRANSCRIPT
KICS North-America Branch Workshop2019.2.9
Hong-Yeop SongYonsei University
Milewski sequences revisited,
and its generalization
Hong-Yeop Song
โข For complex-valued sequences ๐๐,๐๐ of length ๐ฟ๐ฟ, the periodic correlation of ๐๐ and ๐๐ at shift ๐๐ is
๐ช๐ช๐๐,๐๐ ๐๐ = ๏ฟฝ๐๐=๐๐
๐ณ๐ณโ๐๐
๐๐ ๐๐ + ๐๐ ๐๐โ(๐๐)
โซ If ๐๐ is a cyclic shift of ๐๐, it is called autocorrelation, and denoted by ๐ช๐ช๐๐ ๐๐
โซ Otherwise, it is called crosscorrelation
Sequences and Correlation
2
Hong-Yeop Song
Perfect Sequences
โข A sequence ๐๐ of length ๐ณ๐ณ is called perfect if
C๐๐ ๐๐ = ๏ฟฝ๐ฌ๐ฌ, ๐๐ โก 0 mod ๐ฟ๐ฟ0, ๐๐ โข 0 (mod ๐ฟ๐ฟ)
Here, ๐ฌ๐ฌ is called the energy of ๐๐
โข (Sarwate, 79) Crosscorrelation of any two perfect sequences of length ๐ณ๐ณ with the same energy ๐ฌ๐ฌ is lower bounded by ๐ฌ๐ฌ/ ๐ณ๐ณ.โซ An optimal pair of perfect sequences of length ๐ณ๐ณโซ An optimal set of perfect sequences of length ๐ณ๐ณ
3
Hong-Yeop Song
Interleaved Sequence
4
โข Consider two sequences ๐๐0 = ๐๐, ๐๐, ๐๐ and ๐๐1 = ๐๐, ๐๐,๐๐ of length 3 each
โข Write each as a column of an array:
๐๐0, ๐๐1 =๐๐ ๐๐๐๐ ๐๐๐๐ ๐๐
โข Read the array row-by-row and obtain a sequence of length 6:
๐๐ = ๐ผ๐ผ(๐๐๐๐, ๐๐๐๐) = ๐๐,๐๐, ๐๐, ๐๐, ๐๐, ๐๐is called an interleaved sequence of ๐๐0 and ๐๐1
Hong-Yeop Song
History of Perfect Polyphase Sequences
5
1960โs
1970โs
1980โs
1990โs
Frank, ZadoffIRE T-IT, 1962
(Heimiller 1961)
Kumar, Scholtz, WelchJ. Comb. Theory Series
A. 1985
P1 code
Mowโs two unified construction (dissertation 1993, ISSTA 1996)
Alphabet size : ๐๐Period : ๐๐2
Chu(Frank,Zadoff)
IEEE T-IT, 1973
PopovicIEEE T-IT, 1992
P2 codeAlphabet size : ๐๐
Period : ๐๐
Chung, KumarIEEE T-IT, 1989
P4 codeAlphabet size : ๐ ๐ ๐๐
Period : ๐ ๐ ๐๐2
MilewskiIBM J. R&D, 1983
P3 codeAlphabet size : ๐๐๐๐+1
Period : ๐๐2๐๐+1
Hong-Yeop Song 6
The original Milewski constructionLength: ๐๐ โ ๐๐ ๏ฟฝ๐๐๐๐๐๐
MilewskiConstruction
Output perfect polyphase sequence
๐๐ = ๐ ๐ ๐๐ ๐๐=0๐๐๐๐๐๐+๐๐โ1
perfect polyphase Sequence of length ๐๐
๐ท๐ท = ๐ผ๐ผ ๐๐ ๐๐=0๐๐โ1
A positive integer
๐พ๐พ
where
๐๐ ๐๐ = ๐ท๐ท ๐๐ ๐๐๐๐๐๐
๐๐ = ๐๐โ๐๐2๐๐
๐๐1+๐พ๐พ
Here, we use๐๐ = ๐๐๐๐๐๐ + ๐๐ โ (๐๐, ๐๐)
๐๐ ๏ฟฝ ๐๐๐พ๐พ ร ๐๐๐พ๐พ array form of ๐๐
โฎ โฎ โฎโฑ
โฎ โฎ โฎโฑ
โฎ โฎ โฎโฑ
๐ฝ๐ฝ(0) ๐ฝ๐ฝ(0) ๐ฝ๐ฝ(0)๐๐ร ร รโฏ ๐๐๐๐๐ฝ๐ฝ(1) ๐ฝ๐ฝ(1) ๐ฝ๐ฝ(1)๐๐ร ร รโฏ ๐๐๐ต๐ตโ๐๐ ๐๐๐๐
๐ฝ๐ฝ(2) ๐ฝ๐ฝ(2) ๐ฝ๐ฝ(2)๐๐๐๐ร ร รโฏ ๐๐๐ต๐ตโ๐๐ ๐๐๐๐
๐ฝ๐ฝ(๐๐โ 1) ๐ฝ๐ฝ(๐๐โ 1) ๐ฝ๐ฝ(๐๐โ 1)๐๐๐๐โ๐๐ร ร รโฏ ๐๐๐ต๐ตโ๐๐ ๐๐โ๐๐๐๐
๐ฝ๐ฝ(0) ๐ฝ๐ฝ(0) ๐ฝ๐ฝ(0)๐๐๐๐(๐ต๐ตโ๐๐)ร ร รโฏ ๐๐๐ต๐ตโ๐๐ ๐๐(๐ต๐ตโ๐๐)๐๐
๐ฝ๐ฝ(1) ๐ฝ๐ฝ(1) ๐ฝ๐ฝ(1)๐๐๐๐ ๐ต๐ตโ๐๐ +๐๐ร ร รโฏ ๐๐๐ต๐ตโ๐๐ ๐๐ ๐ต๐ตโ๐๐ +๐๐๐๐
๐ฝ๐ฝ(2) ๐ฝ๐ฝ(2) ๐ฝ๐ฝ(2)๐๐๐๐ ๐ต๐ตโ๐๐ +๐๐ร ร รโฏ ๐๐๐ต๐ตโ๐๐ ๐๐ ๐ต๐ตโ๐๐ +๐๐๐๐
๐ฝ๐ฝ(๐๐โ 1) ๐ฝ๐ฝ(๐๐โ 1) ๐ฝ๐ฝ(๐๐โ 1)๐๐๐๐๐ต๐ตโ๐๐ร ร รโฏ ๐๐๐ต๐ตโ๐๐ ๐๐๐ต๐ตโ๐๐๐๐
Input sequence of period ๐๐
๐ต๐ต = ๐๐๐๐
Hong-Yeop Song
Our framework(A special type of interleaved sequences)
7
where
with ๐๐ = ๐๐๐ต๐ต + ๐๐, and ๐๐ = ๐๐๐๐๐๐(โ๐๐๐๐๐๐/๐๐๐ต๐ต).
Interleaving technique
A positive integer๐๐
A polyphase sequenceof length ๐๐
๐๐ ๐๐
A functionโค๐ต๐ต โถ โค๐๐๐ต๐ต
A collection of ๐๐ sequence of length ๐๐๐ฉ๐ฉ = ๐ท๐ท0,๐ท๐ท1, โฆ ,๐ท๐ท๐๐
not necessarily polyphasenot necessarily all distinct
Output sequence ๐ ๐ = ๐๐ ๐๐ ๐๐=0๐๐๐๐2โ1
๐๐ ๐๐ = ๐๐ ๐๐ ๐ท๐ท๐๐ ๐๐ ๐๐๐๐๐๐(๐๐)
Definition. We define A ๐ฉ๐ฉ,๐๐ be a family of interleaved sequences constructed by the above procedure using all possible polyphase sequences ๐๐.
Hong-Yeop Song
Array Form
8
Row index ๐๐
=๐๐,๐๐,๐๐,โฆ
,๐๐๐ต๐ตโ๐๐
Column index ๐๐ = ๐๐,๐๐,๐๐, โฆ ,๐ต๐ตโ ๐๐Assume that ๐๐ is the all-one sequence, โป ๐๐ = ๐๐โ๐๐
2๐๐๐๐๐๐
โฎ โฎ โฎโฑ
โฎ โฎ โฎโฑ
โฎ โฎ โฎโฑ
๐ฝ๐ฝ0(0) ๐ฝ๐ฝ1(0) ๐ฝ๐ฝ๐๐โ1(0)๐๐๐๐(1) ๐๐ร ร รโฏ ๐๐๐๐(๐๐โ1) ๐๐๐๐๐๐(0) 0
๐ฝ๐ฝ0(1) ๐ฝ๐ฝ1(1) ๐ฝ๐ฝ๐๐โ1(1)๐๐๐๐(1) ๐๐ร ร รโฏ ๐๐๐๐(๐๐โ1) ๐๐๐๐๐๐(0) ๐๐
๐ฝ๐ฝ0(2) ๐ฝ๐ฝ1(2) ๐ฝ๐ฝ๐๐โ1(2)๐๐๐๐(1) ๐๐ร ร รโฏ ๐๐๐๐(๐๐โ1) ๐๐๐๐๐๐(0) ๐๐
๐ฝ๐ฝ0(๐๐โ 1) ๐ฝ๐ฝ1(๐๐โ 1) ๐ฝ๐ฝ๐๐โ1(๐๐โ 1)๐๐๐๐(1) ๐๐โ๐๐ร ร รโฏ ๐๐๐๐(๐๐โ1) ๐๐โ๐๐๐๐๐๐(0) (๐๐โ๐๐)
๐ฝ๐ฝ0(0) ๐ฝ๐ฝ1(0) ๐ฝ๐ฝ๐๐โ1(0)๐๐๐๐(1) ๐๐(๐ต๐ตโ๐๐)ร ร รโฏ ๐๐๐๐(๐๐โ1) ๐๐(๐ต๐ตโ๐๐)๐๐๐๐(0) ๐๐(๐ต๐ตโ๐๐)
๐ฝ๐ฝ0(1) ๐ฝ๐ฝ1(1) ๐ฝ๐ฝ๐๐โ1(1)๐๐๐๐(1) ๐๐ ๐ต๐ตโ๐๐ +๐๐ร ร รโฏ ๐๐๐๐(๐๐โ1) ๐๐ ๐ต๐ตโ๐๐ +๐๐๐๐๐๐(0) ๐๐ ๐ต๐ตโ๐๐ +๐๐
๐ฝ๐ฝ0(2) ๐ฝ๐ฝ1(2) ๐ฝ๐ฝ๐๐โ1(2)๐๐๐๐(1) ๐๐ ๐ต๐ตโ๐๐ +๐๐ร ร รโฏ ๐๐๐๐(๐๐โ1) ๐๐ ๐ต๐ตโ๐๐ +๐๐๐๐๐๐(0) ๐๐ ๐ต๐ตโ๐๐ +๐๐
๐ฝ๐ฝ0(๐๐โ 1) ๐ฝ๐ฝ1(๐๐โ 1) ๐ฝ๐ฝ๐๐โ1(๐๐โ 1)๐๐๐๐(1) ๐๐๐ต๐ตโ๐๐ร ร รโฏ ๐๐๐๐(๐๐โ1) ๐๐๐ต๐ตโ๐๐๐๐๐๐(0) ๐๐๐ต๐ตโ๐๐
Input sequence ๐ท๐ท0of period ๐๐
Input sequence ๐ท๐ท1of period ๐๐
Input sequence ๐ท๐ท๐๐โ1of period ๐๐
Input function ๐๐:โค๐ต๐ต โถ โค๐๐๐ต๐ต
Hong-Yeop Song
Milewski Construction is a Special Case
9
MilewskiConstruction
Output perfect polyphase sequence
๐๐ = ๐ ๐ ๐๐ ๐๐=0๐๐๐๐๐๐+๐๐โ1
perfect polyphase sequence
๐ท๐ท = ๐ผ๐ผ ๐๐ ๐๐=0๐๐โ1
A positive integer
๐พ๐พ
where๐ ๐ ๐๐ = ๐ฝ๐ฝ ๐๐ ๐๐๐๐๐๐
with ๐๐ = ๐๐๐๐๐๐ + ๐๐,and
๐๐ = ๐๐โ ๐๐ 2๐๐๐๐1+๐พ๐พ .
Interleaving technique
๐๐ ๐๐
where๐ ๐ ๐๐ = ๐ฝ๐ฝ๐๐ ๐๐ ๐๐๐๐๐๐(๐๐) = ๐ฝ๐ฝ ๐๐ ๐๐๐๐๐๐
with ๐๐ = ๐๐๐ต๐ต + ๐๐ = ๐๐๐๐๐๐ + ๐๐, and
๐๐ = ๐๐โ ๐๐ 2๐๐๐๐๐ต๐ต = ๐๐โ ๐๐ 2๐๐๐๐๐๐+๐๐ .
๐๐๐ต๐ต = ๐ท๐ท0,๐ท๐ท1, โฆ ,๐ท๐ท๐๐
Output sequence
๐๐ = ๐ ๐ ๐๐ ๐๐=0๐๐ ๐๐๐๐ ๐๐
โ1
perfect polyphase sequences
๐ท๐ท0 = ๐ท๐ท1 = โฏ = ๐ท๐ท๐๐โ1An integer๐๐ = ๐๐๐พ๐พ
all-one sequence
The identityfunction๐๐ ๐๐ = ๐๐
=
Hong-Yeop Song
Condition on perfectness(Main result 1)
10
Theorem. Any sequence inA ๐ฉ๐ฉ,๐๐ is perfect if and only if the following conditions are satisfied:
1) ๐ฟ๐ฟ๐๐(๐๐) = ๐๐ for ๐๐ = ๐๐,๐๐, โฆ ,๐ต๐ตโ ๐๐.That is, ๐๐ ๐๐ (๐ฆ๐ฆ๐ฆ๐ฆ๐ฆ๐ฆ ๐ต๐ต) for ๐๐ = ๐๐,๐๐, โฆ ,๐ต๐ตโ ๐๐ is a permutation over โค๐ต๐ต.
2) ๐ฉ๐ฉ is a collection of perfect sequences all of period ๐๐ with the same energy.
We now call themthe generalized Milewski sequences
Definition. Let ๐๐,๐๐ be two functions from โค๐ต๐ต to โค๐๐๐ต๐ต. We define
When ๐๐ = ๐๐, we use ๐ณ๐ณ๐๐ ๐๐ simply.
๐ณ๐ณ๐๐,๐๐ ๐๐ = ๐๐ โ โค๐ต๐ต ๐๐ ๐๐ + ๐๐ โก ๐๐ ๐๐ ๐๐๐๐๐๐ ๐ต๐ต .
Hong-Yeop Song
Examples
11
Constellation of ๐๐
โป ๐๐ = ๐๐โ๐๐2๐๐12
Generalized Milewski
Construction
๐๐ ๐๐๐ต๐ต = ๐ท๐ท0,๐ท๐ท1 ๐๐
๐๐ = 0, 0,โ1,โ๐๐, 1,๐๐2, 0, 0, 1,๐๐4, 1,๐๐5, 0, 0,โ1,โ๐๐7, 1,๐๐8, 0, 0, 1,๐๐10, 1,๐๐11
is a perfect sequence of length 24.
โข ๐ท๐ท0 = ๐ท๐ท1 = 0,โ1,1,0,1,1which is a perfect sequence of length 6,
โข ๐๐ = 2,โข ๐๐(๐๐) = ๐๐, andโข ๐๐ is the all-one sequence.
Hong-Yeop Song
Examples
12โป ๐๐ = ๐๐โ ๐๐2๐๐12
Generalized Milewski
Construction
๐๐ ๐๐๐ต๐ต = ๐ท๐ท0,๐ท๐ท1,๐ท๐ท2 ๐๐
๐๐ is a perfect sequence of length 90.
โข ๐ท๐ท0 = ๐ท๐ท1 = ๐ท๐ท2 =3,โ2,3,โ2,โ2,3,โ2,โ7,โ2,โ2
which is a perfect sequence of period 10โข ๐๐ = 3,โข ๐๐(๐๐) = ๐๐, andโข ๐๐ is the all-one sequence.
Constellation of ๐๐
APSK constellation
ASK constellation
Hong-Yeop Song
Direct vs Indirect
13
Theorem. Assume that ๐๐ is a composite number.1) Any generalized Milewski sequence of length ๐๐๐๐2 from the two-step
method can be also obtained by the direct method.2) There exists a generalized Milewski sequence of length ๐๐๐๐2 from the direct
method which can not be obtained by the two-step method.
Perfect sequences of length ๐๐
Generalized Milewski
sequences of length ๐๐๐๐12
Generalized Milewski
sequences of length ๐๐๐๐2
Two-step synthesis
Direct synthesis
Hong-Yeop Song
Condition on optimal pair(Main result 2)
14
Theorem. Let ๐ต๐ต1 = ๐ท๐ท0,๐ท๐ท1, โฆ ,๐ท๐ท๐๐โ1 and ๐ต๐ต2 = ๐ธ๐ธ0,๐ธ๐ธ1, โฆ ,๐ธ๐ธ๐๐โ1 , all of length ๐๐ and the same energy ๐ธ๐ธ๐ต๐ต, and perfect.
Construct ๐๐ โ A ๐ต๐ต1,๐๐ and ๐๐ โ A(๐ต๐ต2,๐๐).
Then, ๐๐ and ๐๐ have optimal crosscorrelation if and only if the following conditions are satisfied for each ๐๐ = ๐๐,๐๐, โฆ ,๐ต๐ตโ ๐๐:1) ฮจ๐๐,๐๐(๐๐) = 1, i.e., ฮจ๐๐,๐๐ ๐๐ = ๐ฅ๐ฅ .2) For the unique ๐ฅ๐ฅ โ ฮจ๐๐,๐๐(๐๐), the pair of sequences
๐ฝ๐ฝ๐ฅ๐ฅ+๐๐ ๐ก๐ก ๐๐๐๐๐๐ ๐ฅ๐ฅ+๐๐ ๐ก๐ก
๐ก๐ก=0
๐๐โ1and ๐พ๐พ๐ฅ๐ฅ ๐ก๐ก ๐๐๐๐
๐๐ ๐ฅ๐ฅ ๐ก๐ก๐ก๐ก=0
๐๐โ1is optimal.
Hong-Yeop Song
Condition on optimal pair(Simple Special Case)
15
Corollary. Let ๐ต๐ต1 = ๐ท๐ท0,๐ท๐ท1, โฆ ,๐ท๐ท๐๐โ1 and ๐ต๐ต2 = ๐ธ๐ธ0,๐ธ๐ธ1, โฆ ,๐ธ๐ธ๐๐โ1 , all of length ๐๐ and the same energy ๐ธ๐ธ๐ต๐ต, and perfect.
Assume that ๐๐ and ๐๐ have the same range.Construct ๐๐ โ A ๐ต๐ต1,๐๐ and ๐๐ โ A(๐ต๐ต2,๐๐).
Then, ๐๐ and ๐๐ have optimal crosscorrelation if and only if the following conditions are satisfied for each ๐๐ = ๐๐,๐๐, โฆ ,๐ต๐ตโ ๐๐:1) ฮจ๐๐,๐๐(๐๐) = 1, i.e., ฮจ๐๐,๐๐ ๐๐ = ๐ฅ๐ฅ .2) For the unique ๐ฅ๐ฅ โ ฮจ๐๐,๐๐(๐๐),
the pair of sequences ๐ท๐ท๐๐+๐๐ and ๐ธ๐ธ๐๐ is optimal.
Hong-Yeop Song
when ๐๐ = ๐๐โข The all-one sequence of length 1 is a trivial perfect sequence.โข And, we can say that
16
โthe all-one sequence and itself is a (trivial) optimal pair of perfect sequences of length 1โ
an optimal ๐๐-set of generalized Milewski sequences of length ๐ต๐ต๐๐ exists
if and only if a ๐๐ ร ๐๐ circular Florentine array exists
โข Therefore, for ๐๐ = 1,
Hong-Yeop Song
Example
โข For a 4 ร 15 circular Florentine array
we have optimal ๐๐-set of generalized Milewski sequences of length ๐๐2 = 152 by picking up a single perfect sequence from each and every
17
๐๐1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
๐๐2 0 7 1 8 2 12 3 11 9 4 13 5 14 6 10
๐๐3 0 4 11 7 10 1 13 9 5 8 3 6 2 14 12
๐๐4 0 13 7 2 11 6 14 10 3 5 12 9 1 4 8
A 1 ,๐๐1 , A 1 ,๐๐2 , A 1 ,๐๐3 , and
A 1 ,๐๐4 .
Song 00
Hong-Yeop Song
Check๐๐๐๐ ๐๐ + ๐๐ = ๐๐๐๐(๐๐)
has exactly one solution ๐ฅ๐ฅ for any ๐๐
01
2
3
4
5
6
78
9
10
11
12
13
140
7
1
8
2
12
3
119
4
13
5
14
6
10
๐๐1 ๐๐2
01
2
3
4
5
6
78
9
10
11
12
13
14
0 7
1
8
2
12
3
119
4
13
5
14
6
10
๐๐ = 0
01
2
3
4
5
6
78
9
10
11
12
13
14
71
8
2
12
3
11
94
13
5
14
6
10
0
๐๐ = 1
01
2
3
4
5
6
78
9
10
11
12
13
14
18
2
12
3
11
9
4135
14
6
10
07
๐๐ = 2
01
2
3
4
5
6
78
9
10
11
12
13
14
82
12
3
11
9
4
135
14
6
10
0
7
1
๐๐ = 3
01
2
3
4
5
6
78
9
10
11
12
13
14
212
3
11
9
4
13
514
6
10
0
7
1
8
๐๐ = 4
01
2
3
4
5
6
78
9
10
11
12
13
14
123
11
9
4
13
5
14610
0
7
1
8
2
๐๐ = 5
01
2
3
4
5
6
78
9
10
11
12
13
14
๐๐ = 63
11
9
4
13
5
14
6100
7
1
8
2
12
etcโฆ
Hong-Yeop Song
when ๐๐ > ๐๐
โข Construct ๐ ๐ โ A ๐ต๐ต1,๐๐1 , ๐๐ โ A ๐ต๐ต2,๐๐2 with๐ต๐ต1 = ๐ฝ๐ฝ0,๐ฝ๐ฝ1, โฆ ,๐ฝ๐ฝ๐๐โ1 and ๐ต๐ต2 = ๐พ๐พ0, ๐พ๐พ1, โฆ , ๐พ๐พ๐๐โ1 , where
26
๐๐๐๐ 0 1 2 3 4
๐๐๐๐ 0 2 4 1 3
๐ท๐ท0๐ท๐ท1
๐ท๐ท4
โฎ
= ๐ท๐ท= ๐ท๐ท
= ๐ท๐ท
๐ธ๐ธ0๐ธ๐ธ1
๐ธ๐ธ4
โฎ
= ๐ธ๐ธ= ๐ธ๐ธ
= ๐ธ๐ธ
1) ฮจ๐๐,๐๐(๐๐) = 1, i.e., ฮจ๐๐,๐๐ ๐๐ = ๐ฅ๐ฅ .2) For the unique ๐ฅ๐ฅ โ ฮจ๐๐,๐๐(๐๐), the pair of sequences ๐ท๐ท๐๐+๐๐ and ๐ธ๐ธ๐๐ is optimal.
Assume we have an optimal pair ๐ท๐ท,๐ธ๐ธand a ๐๐ ร ๐๐ circular Florentine array:
๐ท๐ท0๐ท๐ท1
๐ท๐ท4
โฎ
= ๐ธ๐ธ= ๐ท๐ท
= ๐ท๐ท
๐ธ๐ธ0๐ธ๐ธ1
๐ธ๐ธ4
โฎ
= ๐ท๐ท= ๐ธ๐ธ
= ๐ธ๐ธ
Then, any ๐ ๐ โ A ๐ต๐ต1,๐๐1and ๐๐ โ A ๐ต๐ต2,๐๐2 is an
optimal pair
Hong-Yeop Song 27
๐๐1 0 1 2 3 4
๐๐2 0 2 4 1 3
Definition. Let ๐๐,๐๐ be two functions from โค๐ต๐ต to โค๐๐๐ต๐ต. ๐ณ๐ณ๐๐,๐๐ ๐๐ = ๐๐ โ โค๐ต๐ต ๐๐ ๐๐ + ๐๐ โก ๐๐ ๐๐ ๐๐๐๐๐๐ ๐ต๐ต .
๐น๐น1,2 ๐๐ = ๐ฅ๐ฅ โ โค๐๐ ๐๐1 ๐ฅ๐ฅ + ๐๐ โก ๐๐2 ๐ฅ๐ฅ ๐๐๐๐๐๐ 5 = 0 โ ๐ฝ๐ฝ0+๐๐ = ๐ฝ๐ฝ๐๐ and ๐พ๐พ๐๐๐น๐น1,2 ๐๐ = ๐ฅ๐ฅ โ โค๐๐ ๐๐1 ๐ฅ๐ฅ + ๐๐ โก ๐๐2 ๐ฅ๐ฅ ๐๐๐๐๐๐ 5 = 2 โ ๐ฝ๐ฝ2+๐๐ = ๐ฝ๐ฝ๐๐ and ๐พ๐พ๐๐๐น๐น1,2 ๐๐ = ๐ฅ๐ฅ โ โค๐๐ ๐๐1 ๐ฅ๐ฅ + ๐๐ โก ๐๐2 ๐ฅ๐ฅ ๐๐๐๐๐๐ 5 = 4 โ ๐ฝ๐ฝ4+๐๐ = ๐ฝ๐ฝ๐๐ and ๐พ๐พ๐๐๐น๐น1,2 ๐๐ = ๐ฅ๐ฅ โ โค๐๐ ๐๐1 ๐ฅ๐ฅ + ๐๐ โก ๐๐2 ๐ฅ๐ฅ ๐๐๐๐๐๐ 5 = 1 โ ๐ฝ๐ฝ1+๐๐ = ๐ฝ๐ฝ๐๐ and ๐พ๐พ๐๐๐น๐น1,2 ๐๐ = ๐ฅ๐ฅ โ โค๐๐ ๐๐1 ๐ฅ๐ฅ + ๐๐ โก ๐๐2 ๐ฅ๐ฅ ๐๐๐๐๐๐ 5 = 3 โ ๐ฝ๐ฝ3+๐๐ = ๐ฝ๐ฝ๐๐ and ๐พ๐พ๐๐
๐ณ๐ณ๐๐,๐๐ ๐๐ = ๐๐ โ โค๐ต๐ต ๐๐๐๐ ๐๐ + ๐๐ โก ๐๐๐๐ ๐๐ ๐๐๐๐๐๐ ๐๐ โ ๐ท๐ท๐๐+๐๐ and ๐ธ๐ธ๐๐
(๐ท๐ท0,
(๐ท๐ท4,
๐ธ๐ธ0) =
๐ธ๐ธ1) =
(๐ท๐ท,๐ธ๐ธ) or (๐ธ๐ธ,๐ท๐ท)
(๐ท๐ท3, ๐ธ๐ธ2) =
(๐ท๐ท,๐ธ๐ธ) or (๐ธ๐ธ,๐ท๐ท)(๐ท๐ท,๐ธ๐ธ) or (๐ธ๐ธ,๐ท๐ท)(๐ท๐ท,๐ธ๐ธ) or (๐ธ๐ธ,๐ท๐ท)(๐ท๐ท,๐ธ๐ธ) or (๐ธ๐ธ,๐ท๐ท)
(๐ท๐ท2, ๐ธ๐ธ3) =(๐ท๐ท1, ๐ธ๐ธ4) =
Hong-Yeop Song
Maximum set size
28
Theorem. Let ๐น๐น๐๐ ๐๐ be the maximal size of circular Florentine arrays with ๐๐columns(symbols). Denote by ๐๐๐บ๐บ(๐๐๐๐2) the maximum size of optimal sets generalized Milewski sequences of length ๐๐๐๐2 from perfect sequences of length ๐๐.1) Assume that ๐๐ = 1. Then
2) Assume that ๐๐ โฅ 2 and let ๐๐๐๐ ๐๐ be the maximum size of optimal perfect sequence sets of period ๐๐. Then,
๐๐๐บ๐บ mN2 = ๐น๐น๐๐ ๐๐ .
๐๐๐บ๐บ = min ๐๐๐๐ ๐๐ ,๐น๐น๐๐ ๐๐ .
Hong-Yeop Song
Maximum set size โ polyphase
29
Corollary. Let ๐๐ = ๐๐๐๐2 be odd and let ๐๐๐๐ ๐ฟ๐ฟ be the maximum size of optimal sets of generalized Milewski polyphase sequences of length ๐ฟ๐ฟ constructed by using Zadoff-Chu sequences of length ๐๐.1) If ๐๐ = 1, then
๐๐๐๐ ๐ฟ๐ฟ = ๐น๐น๐๐ ๐๐ .2) If ๐๐ โฅ 2, then
๐๐๐๐ ๐ฟ๐ฟ = min ๐๐min โ 1,๐น๐น๐๐ ๐๐ ,where ๐๐min โ 1 is the smallest prime factor of ๐๐.
There is no optimal pair of generalized Milewski polyphase sequences of even length constructed by using Zadoff-chu sequences.
(Popovic, 1992) The maximum size of optimal Zadoff-Chu sequence sets with period ๐๐ is ๐๐min โ 1,where ๐๐min is the smallest prime factor of ๐๐.
Hong-Yeop Song
Concluding remarks
โข To obtain an optimal ๐๐-set of generalized Milewski sequences of length ๐๐๐ต๐ต๐๐, we need both:โซ A ๐๐ ร ๐ต๐ต circular Florentine array, andโซ An optimal ๐๐-set of perfect sequences of length ๐๐.
30
3110 x 10 Florentine Square
T. Etzion, S. W. Golomb and H. Taylor,โTuscan-K squares,โAdvances in Applied Mathematics,Vol. 10, pp. 164-174, 1989
H.-Y. Song and J. H. Dinitz, "Tuscan Squares,"CRC Handbook of Combinatorial Designs,edited by C. J. Colbourn and J. H. Dinitz, CRC Press, pp. 480-484, 1996.
H.-Y. Song, โThe existence of circular florentine arrays,โ Comput. Math. Appl., pp. 31-36, June 2000.
0000000000
10 x 11 circular array
Hong-Yeop Song
Concluding remarks
โข To obtain an optimal ๐๐-set of generalized Milewski sequences of length ๐๐๐ต๐ต๐๐, we need both:โซ A ๐๐ ร ๐ต๐ต circular Florentine array, andโซ An optimal ๐๐-set of perfect sequences of length ๐๐.
32
Some open problems:โข For a given integer ๐๐, what is the exact value of ๐น๐น๐๐(๐๐)? โข For a given integer and its smallest prime factor ๐๐min,
is there any other optimal set of size greater then ๐๐min โ 1?
Hong-Yeop Song
Thanks !
33