generalized milewski sequences with perfect
TRANSCRIPT
Generalized Milewski sequenceswith
perfect autocorrelation/optimal crosscorrelation(and its relation with circular Florentine arrays)
ITA 2019, San Diego, California, USAMin Kyu Song, Gangsan Kim and Hong-Yeop Song
Yonsei University, Seoul, Korea
based on the recent submission to IEEE IT Trans in Jan 2019New framework for sequences with perfect autocorrelation and optimal crosscorrelation
Hong-Yeop Song
Contents of Talk
โข Some preliminary concepts
โซ Circular Florentine arrays
โซ Perfect sequencesโซ Interleaved sequencesโซ Milewski construction โ review
โข Proposed generalization
โข Concluding remarks
Hong-Yeop Song
Contents of Talk
โข Some preliminary concepts
โซ Circular Florentine arrays
โซ Perfect sequencesโซ Interleaved sequencesโซ Milewski construction โ review
โข Proposed generalization
โข Concluding remarks
A resurrection of a combinatorial structure after more than 30 years
Circular Florentine arrays
Hong-Yeop Song
Circular Florentine arrays
A ๐๐ ร ๐๐ circular Florentine array is equivalent to a set of ๐๐ distinct permutations ๐๐1,๐๐2, โฆ ,๐๐๐๐ of the integers modulo ๐๐ such that
๐๐๐๐ ๐ฅ๐ฅ + ๐๐ = ๐๐๐๐(๐ฅ๐ฅ)has exactly one solution ๐ฅ๐ฅfor any two distinct permutations ๐๐๐๐, ๐๐๐๐ for any shift ๐๐.
Hong-Yeop Song
Circular Florentine arrays
A ๐๐ ร ๐๐ circular Florentine array is equivalent to a set of ๐๐ distinct permutations ๐๐1,๐๐2, โฆ ,๐๐๐๐ of the integers modulo ๐๐ such that
๐๐๐๐ ๐ฅ๐ฅ + ๐๐ = ๐๐๐๐(๐ฅ๐ฅ)has exactly one solution ๐ฅ๐ฅfor any two distinct permutations ๐๐๐๐, ๐๐๐๐ for any shift ๐๐.
S. W. Golomb and H. Taylor,Tuscan squares โ a new family of combinatorial designs, Ars Combinatoria, 1985.
more than 30 years ago!
Hong-Yeop Song
10ร11 circular Florentine array
0000000000
Hong-Yeop Song
10ร11 circular Tuscan-1 array
0000000000
โข Each circular row is a permutation.
โข For any symbol ๐๐ and distance 1, the symbols from ๐๐ in a circular distance 1 are all distinct
Hong-Yeop Song
10ร11 circular Tuscan-2 array
0000000000
โข Each circular row is a permutation.
โข In addition, for any symbol ๐๐ and distance 2, the symbols from ๐๐ in a circular distance 2 are all distinct
Hong-Yeop Song
10ร11 circular Tuscan-3 array
0000000000
โข Each circular row is a permutation.
โข In addition, for any symbol ๐๐ and distance 3, the symbols from ๐๐ in a circular distance 3 are all distinct
Hong-Yeop Song
10ร11 circular Tuscan-4 array
Hong-Yeop Song
10ร11 circular Tuscan-4 array
10ร11 circular Tuscan-5 array
Hong-Yeop Song
10ร11 circular Tuscan-4 array
10ร11 circular Tuscan-5 array
โฆ10ร11 circular Tuscan-10 array
=10ร11 circular Florentine array
Hong-Yeop Song
10ร11 circular Florentine array
0000000000
โข Each circular row is a permutation.
โข For any symbol ๐๐and each distance d=1,2,โฆ,N-1, the symbols from ๐๐ in a circular distance dare all distinct
Hong-Yeop Song
A transform
0000000000
We may rotateeach row without
violating the property so that
a common symbol comes to the left-most column, and delete the column.
This gives โฆ
Hong-Yeop Song
S. W. Golomb and H. Taylor,Tuscan squares โ a new family of combinatorial designs,Ars Combinatoria, 1985
more than 30 years ago!
10ร10 Florentine square
Hong-Yeop Song
S. W. Golomb and H. Taylor,Tuscan squares โ a new family of combinatorial designs,Ars Combinatoria, 1985
more than 30 years ago!
10ร10 Florentine square
Property of rows
Property of columns, in addition
Hong-Yeop Song
H.-Y. Song and J. H. Dinitz, "Tuscan Squares,"CRC Handbook of Combinatorial Designs, edited by C. J. Colbournand J. H. Dinitz, CRC Press, pp. 480-484, 1996.
S. W. Golomb and H. Taylor,Tuscan squares โ a new family of combinatorial designs,Ars Combinatoria, 1985
more than 30 years ago!
10ร10 Florentine square
Hong-Yeop Song
Circular Florentine arrays
โข Let ๐น๐น๐๐ ๐๐ be the largest integer such that an ๐น๐น๐๐ ๐๐ ร ๐๐circular Florentine array exists. Then,
๐๐min โ 1 โค ๐น๐น๐๐ ๐๐ โค ๐๐ โ 1
.
Hong-Yeop Song
Circular Florentine arrays
โข Let ๐น๐น๐๐ ๐๐ be the largest integer such that an ๐น๐น๐๐ ๐๐ ร ๐๐circular Florentine array exists. Then,
๐๐min โ 1 โค ๐น๐น๐๐ ๐๐ โค ๐๐ โ 1
โข It is known that โซ if ๐๐ is prime, then ๐น๐น๐๐ ๐๐ = ๐๐ โ 1.
Hong-Yeop Song
Circular Florentine arrays
โข Let ๐น๐น๐๐ ๐๐ be the largest integer such that an ๐น๐น๐๐ ๐๐ ร ๐๐circular Florentine array exists. Then,
๐๐min โ 1 โค ๐น๐น๐๐ ๐๐ โค ๐๐ โ 1
โข It is known that โซ if ๐๐ is prime, then ๐น๐น๐๐ ๐๐ = ๐๐ โ 1.โซ Especially, if ๐๐ is even, then ๐น๐น๐๐ ๐๐ = 1.
Hong-Yeop Song
Circular Florentine arrays
โข Let ๐น๐น๐๐ ๐๐ be the largest integer such that an ๐น๐น๐๐ ๐๐ ร ๐๐circular Florentine array exists. Then,
๐๐min โ 1 โค ๐น๐น๐๐ ๐๐ โค ๐๐ โ 1
โข It is known that โซ if ๐๐ is prime, then ๐น๐น๐๐ ๐๐ = ๐๐ โ 1.โซ Especially, if ๐๐ is even, then ๐น๐น๐๐ ๐๐ = 1.โซ For all other odd ๐๐, the exact value ๐น๐น๐๐ ๐๐ is widely
open.
Hong-Yeop Song
A 4 ร 15 circular Florentine array๐๐1 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
Hong-Yeop Song
A 4 ร 15 circular Florentine array
โข 15 = 3 ๏ฟฝ 5 and ๐๐๐๐๐๐๐๐ = 3. Therefore, 2 โค ๐น๐น๐๐ 15 โค 14.โข It turned out that
๐ญ๐ญ๐๐ ๐๐๐๐ = ๐๐and the above example has 4 rows.
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
Hong-Yeop Song
A 4 ร 15 circular Florentine array
โข 15 = 3 ๏ฟฝ 5 and ๐๐๐๐๐๐๐๐ = 3. Therefore, 2 โค ๐น๐น๐๐ 15 โค 14.โข It turned out that
๐ญ๐ญ๐๐ ๐๐๐๐ = ๐๐and the above example has 4 rows.
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
PHD Thesis, USC, by Hong-Yeop Song (1991)and also, later in
Computers & Mathematics with Applications (2000)
Hong-Yeop Song
Check๐ ๐ ๐๐ ๐๐ + ๐๐ = ๐ ๐ ๐๐(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
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)
Sequences and Correlation
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
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
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.
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
Hong-Yeop Song
Interleaved Sequence
โข Consider two sequences ๐๐0 = ๐๐, ๐๐, ๐๐ and ๐๐1 = ๐๐, ๐๐,๐๐ of length 3 each
Hong-Yeop Song
Interleaved Sequence
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
Hong-Yeop Song
Interleaved Sequence
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
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
Hong-Yeop Song
History of Perfect Polyphase Sequences
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
Hong-Yeop Song
History of Perfect Polyphase Sequences
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
MilewskiIBM J. R&D, 1983
P3 codeAlphabet size : ๐๐๐๐+1
Period : ๐๐2๐๐+1
Hong-Yeop Song
History of Perfect Polyphase Sequences
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
The original Milewski constructionLength: ๐๐ โ ๐๐ ๏ฟฝ๐๐๐๐๐๐
Hong-Yeop Song
The original Milewski constructionLength: ๐๐ โ ๐๐ ๏ฟฝ๐๐๐๐๐๐
MilewskiConstruction
perfect polyphase Sequence of length ๐๐
๐ท๐ท = ๐ผ๐ผ ๐๐ ๐๐=0๐๐โ1
A positive integer
๐พ๐พ
Hong-Yeop Song
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)
Hong-Yeop Song
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
Hong-Yeop Song
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 โฎ
โฎ โฎ โฎ
โฎ โฎ โฎ
๐ฝ๐ฝ(0) ๐ฝ๐ฝ(0)โฏ๐ฝ๐ฝ(1) ๐ฝ๐ฝ(1)โฏ
๐ฝ๐ฝ(2) ๐ฝ๐ฝ(2)โฏ
๐ฝ๐ฝ(๐๐โ 1) ๐ฝ๐ฝ(๐๐โ 1) ๐ฝ๐ฝ(๐๐โ 1)โฏ
๐ฝ๐ฝ(0) ๐ฝ๐ฝ(0)โฏ
๐ฝ๐ฝ(1) ๐ฝ๐ฝ(1)โฏ
๐ฝ๐ฝ(2) ๐ฝ๐ฝ(2)โฏ
๐ฝ๐ฝ(๐๐โ 1) ๐ฝ๐ฝ(๐๐โ 1) ๐ฝ๐ฝ(๐๐โ 1)โฏ
Input sequence ๐ท๐ทof period ๐๐
is repeated ๐๐๐พ๐พ times
โฎ
โฎ
โฎ
๐ฝ๐ฝ(0)๐ฝ๐ฝ(1)
๐ฝ๐ฝ(2)
๐ฝ๐ฝ(๐๐โ 1)
๐ฝ๐ฝ(0)
๐ฝ๐ฝ(1)
๐ฝ๐ฝ(2)
๐ฝ๐ฝ(๐๐โ 1)
Hong-Yeop Song
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 = ๐๐โ๐๐๐๐๐ ๐
๐๐๐๐+๐๐
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 ๐ท๐ทof period ๐๐
is repeated ๐๐๐พ๐พ times
Hong-Yeop Song
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 = ๐๐โ๐๐๐๐๐ ๐
๐๐๐๐+๐๐
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
Here, ๐๐ = ๐๐๐พ๐พ and the exponent runs from 0 to ๐๐๐๐ โ 1
Input sequence ๐ท๐ทof period ๐๐
is repeated ๐๐๐พ๐พ times
Hong-Yeop Song
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 = ๐๐โ๐๐๐๐๐ ๐
๐๐๐๐+๐๐
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 ๐ท๐ทof period ๐๐
is repeated ๐๐๐พ๐พ times
the exponent runs from 0 to ๐๐ โ 1
Hong-Yeop Song
Contents of Talk
โข Some preliminary concepts
โซ Circular Florentine arrays
โซ Perfect sequencesโซ Interleaved sequencesโซ Milewski construction โ review
โข Proposed generalization
โข Concluding remarks
A resurrection of a combinatorial structure after more than 30 years
Circular Florentine arrays
Hong-Yeop Song
Our framework(A special type of interleaved sequences)
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
Hong-Yeop Song
Our framework(A special type of interleaved sequences)
where
with ๐๐ = ๐๐๐ต๐ต + ๐๐, and ๐๐ = ๐๐๐๐๐๐(โ๐๐๐๐๐ ๐ /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 ๐๐๐๐๐ ๐ (๐๐)
Hong-Yeop Song
Our framework(A special type of interleaved sequences)
where
with ๐๐ = ๐๐๐ต๐ต + ๐๐, and ๐๐ = ๐๐๐๐๐๐(โ๐๐๐๐๐ ๐ /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 ๐๐๐๐๐ ๐ (๐๐)
Definition. We define A ๐ฉ๐ฉ,๐ ๐ be a family of interleaved sequences constructed by the above procedure using all possible polyphase sequences ๐๐.
Hong-Yeop Song
Array FormAssume that ๐๐ is the all-one sequence, โป ๐๐ = ๐๐โ๐๐
2๐๐๐๐๐๐
Hong-Yeop Song
Array Form
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
Hong-Yeop Song
Array Form
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 โฎ
โฎ โฎ โฎ
โฎ โฎ โฎ
๐ฝ๐ฝ0(0) ๐ฝ๐ฝ1(0) ๐ฝ๐ฝ๐๐โ1(0)โฏ
๐ฝ๐ฝ0(1) ๐ฝ๐ฝ1(1) ๐ฝ๐ฝ๐๐โ1(1)โฏ
๐ฝ๐ฝ0(2) ๐ฝ๐ฝ1(2) ๐ฝ๐ฝ๐๐โ1(2)โฏ
๐ฝ๐ฝ0(๐๐โ 1) ๐ฝ๐ฝ1(๐๐โ 1) ๐ฝ๐ฝ๐๐โ1(๐๐โ 1)โฏ
๐ฝ๐ฝ0(0) ๐ฝ๐ฝ1(0) ๐ฝ๐ฝ๐๐โ1(0)โฏ
๐ฝ๐ฝ0(1) ๐ฝ๐ฝ1(1) ๐ฝ๐ฝ๐๐โ1(1)โฏ
๐ฝ๐ฝ0(2) ๐ฝ๐ฝ1(2) ๐ฝ๐ฝ๐๐โ1(2)โฏ
๐ฝ๐ฝ0(๐๐โ 1) ๐ฝ๐ฝ1(๐๐โ 1) ๐ฝ๐ฝ๐๐โ1(๐๐โ 1)โฏ
Input sequence ๐ท๐ท0of period ๐๐
Repeating ๐๐ times
Input sequence ๐ท๐ท1of period ๐๐
Repeating ๐๐ times
Input sequence ๐ท๐ท๐๐โ1of period ๐๐
Repeating ๐๐ times
Hong-Yeop Song
Array Form
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 function ๐๐:โค๐ต๐ต โถ โค๐๐๐ต๐ต
Hong-Yeop Song
Array Form
Row index ๐๐
=๐๐,๐๐,๐๐,โฆ
,๐๐๐ต๐ตโ๐๐
Column index ๐๐ = ๐๐,๐๐,๐๐, โฆ ,๐ต๐ตโ ๐๐Assume that ๐๐ is the all-one sequence,
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
the exponent runs from ๐๐(0) to ๐๐(๐๐ โ 1)
Hong-Yeop Song
Array Form
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
Here, the exponent of ๐๐๐ ๐ (๐๐)
runs from ๐๐ to ๐๐๐ต๐ตโ ๐๐
Hong-Yeop Song
Milewski Construction is a Special Case
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
Interleaving technique
๐๐ ๐๐ ๐๐๐ต๐ต = ๐ท๐ท0,๐ท๐ท1, โฆ ,๐ท๐ท๐๐
Output sequence
=?
Hong-Yeop Song
Milewski Construction is a Special Case
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
Interleaving technique
๐๐ 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
Milewski Construction is a Special Case
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 = ๐ท๐ท ๐๐ ๐๐๐๐๐๐
Interleaving technique
๐๐ ๐๐
where๐ ๐ ๐๐ = ๐ฝ๐ฝ๐๐ ๐๐ ๐๐๐๐๐ ๐ (๐๐) = ๐ท๐ท ๐๐ ๐๐๐๐๐๐
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
Important Link
Definition. Let ๐ ๐ ,๐๐ be two functions from โค๐ต๐ต to โค๐๐๐ต๐ต. We define
When ๐ ๐ = ๐๐, we use ๐ณ๐ณ๐ ๐ ๐๐ simply.
๐ณ๐ณ๐ ๐ ,๐๐ ๐๐ = ๐๐ โ โค๐ต๐ต ๐ ๐ ๐๐ + ๐๐ โก ๐๐ ๐๐ ๐๐๐๐๐๐ ๐ต๐ต .
Hong-Yeop Song
Condition on perfectness(Main result 1)
Theorem. Any sequence inA ๐ฉ๐ฉ,๐ ๐ is perfect if and only if the following conditions are satisfied:
1) ๐ฟ๐ฟ๐ ๐ (๐๐) = ๐๐ for ๐๐ = ๐๐,๐๐, โฆ ,๐ต๐ตโ ๐๐.That is, ๐ ๐ 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.
Definition. Let ๐ ๐ ,๐๐ be two functions from โค๐ต๐ต to โค๐๐๐ต๐ต. We define
When ๐ ๐ = ๐๐, we use ๐ณ๐ณ๐ ๐ ๐๐ simply.
๐ณ๐ณ๐ ๐ ,๐๐ ๐๐ = ๐๐ โ โค๐ต๐ต ๐ ๐ ๐๐ + ๐๐ โก ๐๐ ๐๐ ๐๐๐๐๐๐ ๐ต๐ต .
Hong-Yeop Song
Condition on perfectness(Main result 1)
Theorem. Any sequence inA ๐ฉ๐ฉ,๐ ๐ is perfect if and only if the following conditions are satisfied:
1) ๐ฟ๐ฟ๐ ๐ (๐๐) = ๐๐ for ๐๐ = ๐๐,๐๐, โฆ ,๐ต๐ตโ ๐๐.That is, ๐ ๐ 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 ๐ ๐ ,๐๐ be two functions from โค๐ต๐ต to โค๐๐๐ต๐ต. We define
When ๐ ๐ = ๐๐, we use ๐ณ๐ณ๐ ๐ ๐๐ simply.
๐ณ๐ณ๐ ๐ ,๐๐ ๐๐ = ๐๐ โ โค๐ต๐ต ๐ ๐ ๐๐ + ๐๐ โก ๐๐ ๐๐ ๐๐๐๐๐๐ ๐ต๐ต .
Hong-Yeop Song
Examples
Generalized Milewski
Construction
๐๐ ๐๐๐ต๐ต = ๐ท๐ท0,๐ท๐ท1 ๐๐
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
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
ExamplesConstellation 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
โป 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.
ASK constellation
Hong-Yeop Song
Examples
โป 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 (when ๐๐ is composite)
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
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
Direct vs Indirect (when ๐๐ is composite)
Hong-Yeop Song
Condition on optimal pair(Main result 2)
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).
Hong-Yeop Song
Condition on optimal pair(Main result 2)
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)
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 ๐ ๐ and ๐๐ have the same range.Construct ๐๐ โ A ๐ต๐ต1,๐๐ and ๐๐ โ A(๐ต๐ต2,๐๐).
Hong-Yeop Song
Condition on optimal pair(Simple Special Case)
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.
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 ๐ ๐ and ๐๐ have the same range.Construct ๐๐ โ A ๐ต๐ต1,๐๐ and ๐๐ โ A(๐ต๐ต2,๐๐).
Hong-Yeop Song
Condition on optimal pair(Simple Special Case)
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.
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 ๐ ๐ and ๐๐ have the same range.Construct ๐๐ โ A ๐ต๐ต1,๐๐ and ๐๐ โ A(๐ต๐ต2,๐๐).
Condition on input pairs
Hong-Yeop Song
Condition on optimal pair(Simple Special Case)
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.
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 ๐ ๐ and ๐๐ have the same range.Construct ๐๐ โ A ๐ต๐ต1,๐๐ and ๐๐ โ A(๐ต๐ต2,๐๐).
Condition on input pairs
Condition on input
permutations
Hong-Yeop Song
when ๐๐ = ๐๐โข The all-one sequence of length 1 is a trivial perfect sequence.โข Therefore,
โthe all-one sequence and itself is a (trivial) optimal pair of perfect sequences of length 1โ
Hong-Yeop Song
when ๐๐ = ๐๐โข The all-one sequence of length 1 is a trivial perfect sequence.โข Therefore,
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, we have:
Hong-Yeop Song
Example
โข For a 4 ร 15 circular Florentine array
we have optimal ๐๐-set of generalized Milewski sequences of length ๐ต๐ต๐๐ = ๐๐๐๐๐๐ by picking up any single perfect sequence from each and every family
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, 91 and 00)
Hong-Yeop Song
Example
โข For a ๐๐ ร ๐๐๐๐ circular Florentine array
we have optimal ๐๐-set of generalized Milewski sequences of length ๐ต๐ต๐๐ = ๐๐๐๐๐๐ by picking up any single perfect sequence from each and every family
โข New, in the sense of size 4 (previously known only of size 2) for length ๐๐๐๐๐๐ or ๐๐๐๐
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, 91 and 00)
Hong-Yeop Song
when ๐๐ > ๐๐
๐ ๐ ๐๐ 0 1 2 3 4
๐ ๐ ๐๐ 0 2 4 1 3Assume we have an optimal pair ๐ท๐ท,๐ธ๐ธand a ๐๐ ร ๐๐ circular Florentine array:
Hong-Yeop Song
when ๐๐ > ๐๐
๐ ๐ ๐๐ 0 1 2 3 4
๐ ๐ ๐๐ 0 2 4 1 3
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:
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
๐ ๐ ๐๐ 0 1 2 3 4
๐ ๐ ๐๐ 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:
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
๐ ๐ ๐๐ 0 1 2 3 4
๐ ๐ ๐๐ 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:
Then, any ๐ ๐ โ A ๐ต๐ต1,๐๐1and ๐๐ โ A ๐ต๐ต2,๐๐2 is an
optimal pair
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
๐ ๐ ๐๐ 0 1 2 3 4
๐ ๐ ๐๐ 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
๐๐1 0 1 2 3 4
๐๐2 0 2 4 1 3
Definition. Let ๐ ๐ ,๐๐ be two functions from โค๐ต๐ต to โค๐๐๐ต๐ต. ๐ณ๐ณ๐ ๐ ,๐๐ ๐๐ = ๐๐ โ โค๐ต๐ต ๐ ๐ ๐๐ + ๐๐ โก ๐๐ ๐๐ ๐๐๐๐๐๐ ๐ต๐ต .
๐ณ๐ณ๐๐,๐๐ ๐๐ = ๐๐ โ โค๐ต๐ต ๐ ๐ ๐๐ ๐๐ + ๐๐ โก ๐ ๐ ๐๐ ๐๐ ๐๐๐๐๐๐ ๐๐ โ ๐ท๐ท๐๐+๐๐ and ๐ธ๐ธ๐๐
Hong-Yeop Song
๐๐1 0 1 2 3 4
๐๐2 0 2 4 1 3
Definition. Let ๐ ๐ ,๐๐ be two functions from โค๐ต๐ต to โค๐๐๐ต๐ต. ๐ณ๐ณ๐ ๐ ,๐๐ ๐๐ = ๐๐ โ โค๐ต๐ต ๐ ๐ 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 ๐ ๐ ๐๐ ๐๐ + ๐๐ โก ๐ ๐ ๐๐ ๐๐ ๐๐๐๐๐๐ ๐๐ โ ๐ท๐ท๐๐+๐๐ and ๐ธ๐ธ๐๐
Hong-Yeop Song
๐๐1 0 1 2 3 4
๐๐2 0 2 4 1 3
Definition. Let ๐ ๐ ,๐๐ be two functions from โค๐ต๐ต to โค๐๐๐ต๐ต. ๐ณ๐ณ๐ ๐ ,๐๐ ๐๐ = ๐๐ โ โค๐ต๐ต ๐ ๐ 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 ๐ ๐ ๐๐ ๐๐ + ๐๐ โก ๐ ๐ ๐๐ 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
โข Let ๐ญ๐ญ๐๐ ๐ต๐ต be the maximum such that an ๐ญ๐ญ๐๐ ๐ต๐ต ร ๐ต๐ตcircular Florentine array exists.
โข Let ๐ถ๐ถ๐ฎ๐ฎ(๐๐๐ต๐ต๐๐) be the maximum such that an optimal ๐ถ๐ถ๐ฎ๐ฎ(๐๐๐ต๐ต๐๐)-set of generalized Milewski sequences of length ๐๐๐ต๐ต๐๐ from perfect sequences of length ๐๐.
Hong-Yeop Song
Maximum set size
โข Let ๐ญ๐ญ๐๐ ๐ต๐ต be the maximum such that an ๐ญ๐ญ๐๐ ๐ต๐ต ร ๐ต๐ตcircular Florentine array exists.
โข Let ๐ถ๐ถ๐ฎ๐ฎ(๐๐๐ต๐ต๐๐) be the maximum such that an optimal ๐ถ๐ถ๐ฎ๐ฎ(๐๐๐ต๐ต๐๐)-set of generalized Milewski sequences of length ๐๐๐ต๐ต๐๐ from perfect sequences of length ๐๐.
1) When ๐๐ = 1,๐๐๐บ๐บ mN2 = ๐๐๐บ๐บ N2 = ๐น๐น๐๐ ๐๐ .
Hong-Yeop Song
Maximum set size
โข Let ๐ญ๐ญ๐๐ ๐ต๐ต be the maximum such that an ๐ญ๐ญ๐๐ ๐ต๐ต ร ๐ต๐ตcircular Florentine array exists.
โข Let ๐ถ๐ถ๐ฎ๐ฎ(๐๐๐ต๐ต๐๐) be the maximum such that an optimal ๐ถ๐ถ๐ฎ๐ฎ(๐๐๐ต๐ต๐๐)-set of generalized Milewski sequences of length ๐๐๐ต๐ต๐๐ from perfect sequences of length ๐๐.
1) When ๐๐ = 1,๐๐๐บ๐บ mN2 = ๐๐๐บ๐บ N2 = ๐น๐น๐๐ ๐๐ .
2) When ๐๐ โฅ 2, ๐๐๐บ๐บ mN2 = min ๐๐๐๐ ๐๐ ,๐น๐น๐๐ ๐๐
where ๐๐๐๐ ๐๐ is the maximum such that an optimal ๐๐๐๐ ๐๐ -set of perfect sequences of period 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.
(When ๐๐ = 1, a trivial example will work always)
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.
(When ๐๐ = 1, a trivial example will work always)
Some open problems:โข Find any other positive odd integer ๐ต๐ต such that ๐ญ๐ญ๐๐(๐ต๐ต)
is greater than ๐๐๐๐๐๐๐๐ โ ๐๐.โข Determine the exact value of ๐ญ๐ญ๐๐(๐ต๐ต) for every odd ๐ต๐ต.
Hong-Yeop Song
Thanks !