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

• Some preliminary concepts

▫ Circular Florentine arrays

▫ Perfect sequences▫ Interleaved sequences▫ Milewski construction – review

• Proposed generalization

• Concluding remarks

Hong-Yeop Song

Contents of Talk

• Some preliminary concepts

▫ Circular Florentine arrays

▫ Perfect sequences▫ Interleaved sequences▫ Milewski construction – review

• Proposed generalization

• Concluding remarks

A resurrection of a combinatorial structure after more than 30 years

Circular Florentine arrays

Hong-Yeop Song

Circular Florentine arrays

A 𝑘𝑘 × 𝑁𝑁 circular Florentine array is equivalent to a set of 𝑘𝑘 distinct permutations 𝜋𝜋1,𝜋𝜋2, … ,𝜋𝜋𝑘𝑘 of the integers modulo 𝑁𝑁 such that

𝜋𝜋𝑖𝑖 𝑥𝑥 + 𝜏𝜏 = 𝜋𝜋𝑗𝑗(𝑥𝑥)has exactly one solution 𝑥𝑥for any two distinct permutations 𝜋𝜋𝑖𝑖, 𝜋𝜋𝑗𝑗 for any shift 𝜏𝜏.

Hong-Yeop Song

Circular Florentine arrays

A 𝑘𝑘 × 𝑁𝑁 circular Florentine array is equivalent to a set of 𝑘𝑘 distinct permutations 𝜋𝜋1,𝜋𝜋2, … ,𝜋𝜋𝑘𝑘 of the integers modulo 𝑁𝑁 such that

𝜋𝜋𝑖𝑖 𝑥𝑥 + 𝜏𝜏 = 𝜋𝜋𝑗𝑗(𝑥𝑥)has exactly one solution 𝑥𝑥for any two distinct permutations 𝜋𝜋𝑖𝑖, 𝜋𝜋𝑗𝑗 for any shift 𝜏𝜏.

S. W. Golomb and H. Taylor,Tuscan squares – a new family of combinatorial designs, Ars Combinatoria, 1985.

more than 30 years ago!

Hong-Yeop Song

10×11 circular Florentine array

0000000000

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10×11 circular Tuscan-1 array

0000000000

• Each circular row is a permutation.

• For any symbol 𝒂𝒂 and distance 1, the symbols from 𝒂𝒂 in a circular distance 1 are all distinct

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10×11 circular Tuscan-2 array

0000000000

• Each circular row is a permutation.

• In addition, for any symbol 𝒂𝒂 and distance 2, the symbols from 𝒂𝒂 in a circular distance 2 are all distinct

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10×11 circular Tuscan-3 array

0000000000

• Each circular row is a permutation.

• In addition, for any symbol 𝒂𝒂 and distance 3, the symbols from 𝒂𝒂 in a circular distance 3 are all distinct

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10×11 circular Tuscan-4 array

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10×11 circular Tuscan-4 array

10×11 circular Tuscan-5 array

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10×11 circular Tuscan-4 array

10×11 circular Tuscan-5 array

…10×11 circular Tuscan-10 array

=10×11 circular Florentine array

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10×11 circular Florentine array

0000000000

• Each circular row is a permutation.

• For any symbol 𝒂𝒂and each distance d=1,2,…,N-1, the symbols from 𝒂𝒂 in a circular distance dare all distinct

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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 …

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S. W. Golomb and H. Taylor,Tuscan squares – a new family of combinatorial designs,Ars Combinatoria, 1985

more than 30 years ago!

10×10 Florentine square

Hong-Yeop Song

S. W. Golomb and H. Taylor,Tuscan squares – a new family of combinatorial designs,Ars Combinatoria, 1985

more than 30 years ago!

10×10 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 – a new family of combinatorial designs,Ars Combinatoria, 1985

more than 30 years ago!

10×10 Florentine square

Hong-Yeop Song

Circular Florentine arrays

• Let 𝐹𝐹𝑐𝑐 𝑁𝑁 be the largest integer such that an 𝐹𝐹𝑐𝑐 𝑁𝑁 × 𝑁𝑁circular Florentine array exists. Then,

𝑝𝑝min − 1 ≤ 𝐹𝐹𝑐𝑐 𝑁𝑁 ≤ 𝑁𝑁 − 1

.

Hong-Yeop Song

Circular Florentine arrays

• Let 𝐹𝐹𝑐𝑐 𝑁𝑁 be the largest integer such that an 𝐹𝐹𝑐𝑐 𝑁𝑁 × 𝑁𝑁circular Florentine array exists. Then,

𝑝𝑝min − 1 ≤ 𝐹𝐹𝑐𝑐 𝑁𝑁 ≤ 𝑁𝑁 − 1

• It is known that ▫ if 𝑁𝑁 is prime, then 𝐹𝐹𝑐𝑐 𝑁𝑁 = 𝑁𝑁 − 1.

Hong-Yeop Song

Circular Florentine arrays

• Let 𝐹𝐹𝑐𝑐 𝑁𝑁 be the largest integer such that an 𝐹𝐹𝑐𝑐 𝑁𝑁 × 𝑁𝑁circular Florentine array exists. Then,

𝑝𝑝min − 1 ≤ 𝐹𝐹𝑐𝑐 𝑁𝑁 ≤ 𝑁𝑁 − 1

• It is known that ▫ if 𝑁𝑁 is prime, then 𝐹𝐹𝑐𝑐 𝑁𝑁 = 𝑁𝑁 − 1.▫ Especially, if 𝑁𝑁 is even, then 𝐹𝐹𝑐𝑐 𝑁𝑁 = 1.

Hong-Yeop Song

Circular Florentine arrays

• Let 𝐹𝐹𝑐𝑐 𝑁𝑁 be the largest integer such that an 𝐹𝐹𝑐𝑐 𝑁𝑁 × 𝑁𝑁circular Florentine array exists. Then,

𝑝𝑝min − 1 ≤ 𝐹𝐹𝑐𝑐 𝑁𝑁 ≤ 𝑁𝑁 − 1

• It is known that ▫ if 𝑁𝑁 is prime, then 𝐹𝐹𝑐𝑐 𝑁𝑁 = 𝑁𝑁 − 1.▫ Especially, if 𝑁𝑁 is even, then 𝐹𝐹𝑐𝑐 𝑁𝑁 = 1.▫ For all other odd 𝑁𝑁, the exact value 𝐹𝐹𝑐𝑐 𝑁𝑁 is widely

open.

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A 4 × 15 circular Florentine array𝜋𝜋1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

𝜋𝜋2 0 7 1 8 2 12 3 11 9 4 13 5 14 6 10

𝜋𝜋3 0 4 11 7 10 1 13 9 5 8 3 6 2 14 12

𝜋𝜋4 0 13 7 2 11 6 14 10 3 5 12 9 1 4 8

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A 4 × 15 circular Florentine array

• 15 = 3 � 5 and 𝑝𝑝𝑚𝑚𝑖𝑖𝑚𝑚 = 3. Therefore, 2 ≤ 𝐹𝐹𝑐𝑐 15 ≤ 14.• It turned out that

𝑭𝑭𝒄𝒄 𝟏𝟏𝟏𝟏 = 𝟒𝟒and the above example has 4 rows.

𝜋𝜋1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

𝜋𝜋2 0 7 1 8 2 12 3 11 9 4 13 5 14 6 10

𝜋𝜋3 0 4 11 7 10 1 13 9 5 8 3 6 2 14 12

𝜋𝜋4 0 13 7 2 11 6 14 10 3 5 12 9 1 4 8

Hong-Yeop Song

A 4 × 15 circular Florentine array

• 15 = 3 � 5 and 𝑝𝑝𝑚𝑚𝑖𝑖𝑚𝑚 = 3. Therefore, 2 ≤ 𝐹𝐹𝑐𝑐 15 ≤ 14.• It turned out that

𝑭𝑭𝒄𝒄 𝟏𝟏𝟏𝟏 = 𝟒𝟒and the above example has 4 rows.

𝜋𝜋1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

𝜋𝜋2 0 7 1 8 2 12 3 11 9 4 13 5 14 6 10

𝜋𝜋3 0 4 11 7 10 1 13 9 5 8 3 6 2 14 12

𝜋𝜋4 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

Check𝝅𝝅𝟐𝟐 𝒙𝒙 + 𝝉𝝉 = 𝝅𝝅𝟏𝟏(𝒙𝒙)

has exactly one solution 𝑥𝑥 for any 𝜏𝜏

01

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78

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140

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119

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14

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10

𝜋𝜋1 𝜋𝜋2

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78

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𝜏𝜏 = 0

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𝜏𝜏 = 1

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4135

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𝜏𝜏 = 2

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82

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135

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𝜏𝜏 = 3

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212

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514

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𝜏𝜏 = 4

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𝜏𝜏 = 5

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𝜏𝜏 = 63

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6100

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12

etc…

Hong-Yeop Song

• For complex-valued sequences 𝒙𝒙,𝒚𝒚 of length 𝐿𝐿, the periodic correlation of 𝒙𝒙 and 𝒚𝒚 at shift 𝝉𝝉 is

𝑪𝑪𝒙𝒙,𝒚𝒚 𝝉𝝉 = �𝒏𝒏=𝟎𝟎

𝑳𝑳−𝟏𝟏

𝒙𝒙 𝒏𝒏 + 𝝉𝝉 𝒚𝒚∗(𝒏𝒏)

Sequences and Correlation

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• For complex-valued sequences 𝒙𝒙,𝒚𝒚 of length 𝐿𝐿, the periodic correlation of 𝒙𝒙 and 𝒚𝒚 at shift 𝝉𝝉 is

𝑪𝑪𝒙𝒙,𝒚𝒚 𝝉𝝉 = �𝒏𝒏=𝟎𝟎

𝑳𝑳−𝟏𝟏

𝒙𝒙 𝒏𝒏 + 𝝉𝝉 𝒚𝒚∗(𝒏𝒏)

▫ If 𝒚𝒚 is a cyclic shift of 𝒙𝒙, it is called autocorrelation, and denoted by 𝑪𝑪𝒙𝒙 𝝉𝝉

▫ Otherwise, it is called crosscorrelation

Sequences and Correlation

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Perfect Sequences

• A sequence 𝒙𝒙 of length 𝑳𝑳 is called perfect if

C𝒙𝒙 𝜏𝜏 = �𝑬𝑬, 𝜏𝜏 ≡ 0 mod 𝐿𝐿0, 𝜏𝜏 ≢ 0 (mod 𝐿𝐿)

Here, 𝑬𝑬 is called the energy of 𝒙𝒙

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Perfect Sequences

• A sequence 𝒙𝒙 of length 𝑳𝑳 is called perfect if

C𝒙𝒙 𝜏𝜏 = �𝑬𝑬, 𝜏𝜏 ≡ 0 mod 𝐿𝐿0, 𝜏𝜏 ≢ 0 (mod 𝐿𝐿)

Here, 𝑬𝑬 is called the energy of 𝒙𝒙

• (Sarwate, 79) Crosscorrelation of any two perfect sequences of length 𝑳𝑳 with the same energy 𝑬𝑬 is lower bounded by 𝑬𝑬/ 𝑳𝑳.

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Perfect Sequences

• A sequence 𝒙𝒙 of length 𝑳𝑳 is called perfect if

C𝒙𝒙 𝜏𝜏 = �𝑬𝑬, 𝜏𝜏 ≡ 0 mod 𝐿𝐿0, 𝜏𝜏 ≢ 0 (mod 𝐿𝐿)

Here, 𝑬𝑬 is called the energy of 𝒙𝒙

• (Sarwate, 79) Crosscorrelation of any two perfect sequences of length 𝑳𝑳 with the same energy 𝑬𝑬 is lower bounded by 𝑬𝑬/ 𝑳𝑳.▫ An optimal pair of perfect sequences of length 𝑳𝑳▫ An optimal set of perfect sequences of length 𝑳𝑳

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Interleaved Sequence

• Consider two sequences 𝒔𝒔0 = 𝑎𝑎, 𝑏𝑏, 𝑐𝑐 and 𝒔𝒔1 = 𝑑𝑑, 𝑒𝑒,𝑓𝑓 of length 3 each

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Interleaved Sequence

• Consider two sequences 𝒔𝒔0 = 𝑎𝑎, 𝑏𝑏, 𝑐𝑐 and 𝒔𝒔1 = 𝑑𝑑, 𝑒𝑒,𝑓𝑓 of length 3 each

• Write each as a column of an array:

𝒔𝒔0, 𝒔𝒔1 =𝑎𝑎 𝑑𝑑𝑏𝑏 𝑒𝑒𝑐𝑐 𝑓𝑓

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Interleaved Sequence

• Consider two sequences 𝒔𝒔0 = 𝑎𝑎, 𝑏𝑏, 𝑐𝑐 and 𝒔𝒔1 = 𝑑𝑑, 𝑒𝑒,𝑓𝑓 of length 3 each

• Write each as a column of an array:

𝒔𝒔0, 𝒔𝒔1 =𝑎𝑎 𝑑𝑑𝑏𝑏 𝑒𝑒𝑐𝑐 𝑓𝑓

• Read the array row-by-row and obtain a sequence of length 6:

𝒔𝒔 = 𝐼𝐼(𝒔𝒔𝟎𝟎, 𝒔𝒔𝟏𝟏) = 𝑎𝑎,𝑑𝑑, 𝑏𝑏, 𝑒𝑒, 𝑐𝑐, 𝑓𝑓is called an interleaved sequence of 𝒔𝒔0 and 𝒔𝒔1

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History of Perfect Polyphase Sequences

1960’s

1970’s

1980’s

1990’s

Frank, ZadoffIRE T-IT, 1962

(Heimiller 1961)

Kumar, Scholtz, WelchJ. Comb. Theory Series

A. 1985

P1 code

Mow’s two unified construction (dissertation 1993, ISSTA 1996)

Alphabet size : 𝑁𝑁Period : 𝑁𝑁2

Hong-Yeop Song

History of Perfect Polyphase Sequences

1960’s

1970’s

1980’s

1990’s

Frank, ZadoffIRE T-IT, 1962

(Heimiller 1961)

Kumar, Scholtz, WelchJ. Comb. Theory Series

A. 1985

P1 code

Mow’s two unified construction (dissertation 1993, ISSTA 1996)

Alphabet size : 𝑁𝑁Period : 𝑁𝑁2

Chu(Frank,Zadoff)

IEEE T-IT, 1973

PopovicIEEE T-IT, 1992

P2 codeAlphabet size : 𝑁𝑁

Period : 𝑁𝑁

Hong-Yeop Song

History of Perfect Polyphase Sequences

1960’s

1970’s

1980’s

1990’s

Frank, ZadoffIRE T-IT, 1962

(Heimiller 1961)

Kumar, Scholtz, WelchJ. Comb. Theory Series

A. 1985

P1 code

Mow’s two unified construction (dissertation 1993, ISSTA 1996)

Alphabet size : 𝑁𝑁Period : 𝑁𝑁2

Chu(Frank,Zadoff)

IEEE T-IT, 1973

PopovicIEEE T-IT, 1992

P2 codeAlphabet size : 𝑁𝑁

Period : 𝑁𝑁

MilewskiIBM J. R&D, 1983

P3 codeAlphabet size : 𝑁𝑁𝑘𝑘+1

Period : 𝑁𝑁2𝑘𝑘+1

Hong-Yeop Song

History of Perfect Polyphase Sequences

1960’s

1970’s

1980’s

1990’s

Frank, ZadoffIRE T-IT, 1962

(Heimiller 1961)

Kumar, Scholtz, WelchJ. Comb. Theory Series

A. 1985

P1 code

Mow’s two unified construction (dissertation 1993, ISSTA 1996)

Alphabet size : 𝑁𝑁Period : 𝑁𝑁2

Chu(Frank,Zadoff)

IEEE T-IT, 1973

PopovicIEEE T-IT, 1992

P2 codeAlphabet size : 𝑁𝑁

Period : 𝑁𝑁

Chung, KumarIEEE T-IT, 1989

P4 codeAlphabet size : 𝑠𝑠𝑁𝑁

Period : 𝑠𝑠𝑁𝑁2

MilewskiIBM J. R&D, 1983

P3 codeAlphabet size : 𝑁𝑁𝑘𝑘+1

Period : 𝑁𝑁2𝑘𝑘+1

Hong-Yeop Song

The original Milewski constructionLength: 𝒎𝒎 → 𝒎𝒎 �𝒎𝒎𝟐𝟐𝟐𝟐

Hong-Yeop Song

The original Milewski constructionLength: 𝒎𝒎 → 𝒎𝒎 �𝒎𝒎𝟐𝟐𝟐𝟐

MilewskiConstruction

perfect polyphase Sequence of length 𝑚𝑚

𝜷𝜷 = 𝛼𝛼 𝑛𝑛 𝑚𝑚=0𝑚𝑚−1

A positive integer

𝐾𝐾

Hong-Yeop Song

The original Milewski constructionLength: 𝒎𝒎 → 𝒎𝒎 �𝒎𝒎𝟐𝟐𝟐𝟐

MilewskiConstruction

Output perfect polyphase sequence

𝒔𝒔 = 𝑠𝑠 𝑛𝑛 𝑚𝑚=0𝒎𝒎𝟐𝟐𝟐𝟐+𝟏𝟏−1

perfect polyphase Sequence of length 𝑚𝑚

𝜷𝜷 = 𝛼𝛼 𝑛𝑛 𝑚𝑚=0𝑚𝑚−1

A positive integer

𝐾𝐾

where

𝒔𝒔 𝒏𝒏 = 𝜷𝜷 𝒒𝒒 𝝎𝝎𝒒𝒒𝒓𝒓

𝜔𝜔 = 𝑒𝑒−𝑗𝑗2𝜋𝜋

𝑚𝑚1+𝐾𝐾

Here, we use𝑛𝑛 = 𝒒𝒒𝒎𝒎𝟐𝟐 + 𝒓𝒓 ↔ (𝒒𝒒, 𝒓𝒓)

Hong-Yeop Song

The original Milewski constructionLength: 𝒎𝒎 → 𝒎𝒎 �𝒎𝒎𝟐𝟐𝟐𝟐

MilewskiConstruction

Output perfect polyphase sequence

𝒔𝒔 = 𝑠𝑠 𝑛𝑛 𝑚𝑚=0𝒎𝒎𝟐𝟐𝟐𝟐+𝟏𝟏−1

perfect polyphase Sequence of length 𝑚𝑚

𝜷𝜷 = 𝛼𝛼 𝑛𝑛 𝑚𝑚=0𝑚𝑚−1

A positive integer

𝐾𝐾

where

𝒔𝒔 𝒏𝒏 = 𝜷𝜷 𝒒𝒒 𝝎𝝎𝒒𝒒𝒓𝒓

𝜔𝜔 = 𝑒𝑒−𝑗𝑗2𝜋𝜋

𝑚𝑚1+𝐾𝐾

Here, we use𝑛𝑛 = 𝒒𝒒𝒎𝒎𝟐𝟐 + 𝒓𝒓 ↔ (𝒒𝒒, 𝒓𝒓)

𝑚𝑚 � 𝑚𝑚𝐾𝐾 × 𝑚𝑚𝐾𝐾 array form of 𝒔𝒔

Hong-Yeop Song

The original Milewski constructionLength: 𝒎𝒎 → 𝒎𝒎 �𝒎𝒎𝟐𝟐𝟐𝟐

MilewskiConstruction

Output perfect polyphase sequence

𝒔𝒔 = 𝑠𝑠 𝑛𝑛 𝑚𝑚=0𝒎𝒎𝟐𝟐𝟐𝟐+𝟏𝟏−1

perfect polyphase Sequence of length 𝑚𝑚

𝜷𝜷 = 𝛼𝛼 𝑛𝑛 𝑚𝑚=0𝑚𝑚−1

A positive integer

𝐾𝐾

where

𝒔𝒔 𝒏𝒏 = 𝜷𝜷 𝒒𝒒 𝝎𝝎𝒒𝒒𝒓𝒓

𝜔𝜔 = 𝑒𝑒−𝑗𝑗2𝜋𝜋

𝑚𝑚1+𝐾𝐾

Here, we use𝑛𝑛 = 𝒒𝒒𝒎𝒎𝟐𝟐 + 𝒓𝒓 ↔ (𝒒𝒒, 𝒓𝒓)

𝑚𝑚 � 𝑚𝑚𝐾𝐾 × 𝑚𝑚𝐾𝐾 array form of 𝒔𝒔

⋮ ⋮ ⋮

⋮ ⋮ ⋮

⋮ ⋮ ⋮

𝛽𝛽(0) 𝛽𝛽(0)⋯𝛽𝛽(1) 𝛽𝛽(1)⋯

𝛽𝛽(2) 𝛽𝛽(2)⋯

𝛽𝛽(𝑚𝑚− 1) 𝛽𝛽(𝑚𝑚− 1) 𝛽𝛽(𝑚𝑚− 1)⋯

𝛽𝛽(0) 𝛽𝛽(0)⋯

𝛽𝛽(1) 𝛽𝛽(1)⋯

𝛽𝛽(2) 𝛽𝛽(2)⋯

𝛽𝛽(𝑚𝑚− 1) 𝛽𝛽(𝑚𝑚− 1) 𝛽𝛽(𝑚𝑚− 1)⋯

Input sequence 𝜷𝜷of period 𝑚𝑚

is repeated 𝑚𝑚𝐾𝐾 times

⋮

⋮

⋮

𝛽𝛽(0)𝛽𝛽(1)

𝛽𝛽(2)

𝛽𝛽(𝑚𝑚− 1)

𝛽𝛽(0)

𝛽𝛽(1)

𝛽𝛽(2)

𝛽𝛽(𝑚𝑚− 1)

Hong-Yeop Song

The original Milewski constructionLength: 𝒎𝒎 → 𝒎𝒎 �𝒎𝒎𝟐𝟐𝟐𝟐

MilewskiConstruction

Output perfect polyphase sequence

𝒔𝒔 = 𝑠𝑠 𝑛𝑛 𝑚𝑚=0𝒎𝒎𝟐𝟐𝟐𝟐+𝟏𝟏−1

perfect polyphase Sequence of length 𝑚𝑚

𝜷𝜷 = 𝛼𝛼 𝑛𝑛 𝑚𝑚=0𝑚𝑚−1

A positive integer

𝐾𝐾

where

𝒔𝒔 𝒏𝒏 = 𝜷𝜷 𝒒𝒒 𝝎𝝎𝒒𝒒𝒓𝒓

𝝎𝝎 = 𝒆𝒆−𝒋𝒋𝟐𝟐𝝅𝝅

𝒎𝒎𝟏𝟏+𝟐𝟐

Here, we use𝑛𝑛 = 𝒒𝒒𝒎𝒎𝟐𝟐 + 𝒓𝒓 ↔ (𝒒𝒒, 𝒓𝒓)

𝑚𝑚 � 𝑚𝑚𝐾𝐾 × 𝑚𝑚𝐾𝐾 array form of 𝒔𝒔

⋮ ⋮ ⋮⋱

⋮ ⋮ ⋮⋱

⋮ ⋮ ⋮⋱

𝛽𝛽(0) 𝛽𝛽(0) 𝛽𝛽(0)𝟏𝟏× × ×⋯ 𝟏𝟏𝟏𝟏𝛽𝛽(1) 𝛽𝛽(1) 𝛽𝛽(1)𝝎𝝎× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝟏𝟏𝟏𝟏

𝛽𝛽(2) 𝛽𝛽(2) 𝛽𝛽(2)𝝎𝝎𝟐𝟐× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝟐𝟐𝟏𝟏

𝛽𝛽(𝑚𝑚− 1) 𝛽𝛽(𝑚𝑚− 1) 𝛽𝛽(𝑚𝑚− 1)𝝎𝝎𝒎𝒎−𝟏𝟏× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝒎𝒎−𝟏𝟏𝟏𝟏

𝛽𝛽(0) 𝛽𝛽(0) 𝛽𝛽(0)𝝎𝝎𝒎𝒎(𝑵𝑵−𝟏𝟏)× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝒎𝒎(𝑵𝑵−𝟏𝟏)𝟏𝟏

𝛽𝛽(1) 𝛽𝛽(1) 𝛽𝛽(1)𝝎𝝎𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟏𝟏× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟏𝟏𝟏𝟏

𝛽𝛽(2) 𝛽𝛽(2) 𝛽𝛽(2)𝝎𝝎𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟐𝟐× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟐𝟐𝟏𝟏

𝛽𝛽(𝑚𝑚− 1) 𝛽𝛽(𝑚𝑚− 1) 𝛽𝛽(𝑚𝑚− 1)𝝎𝝎𝒎𝒎𝑵𝑵−𝟏𝟏× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝒎𝒎𝑵𝑵−𝟏𝟏𝟏𝟏

Input sequence 𝜷𝜷of period 𝑚𝑚

is repeated 𝑚𝑚𝐾𝐾 times

Hong-Yeop Song

The original Milewski constructionLength: 𝒎𝒎 → 𝒎𝒎 �𝒎𝒎𝟐𝟐𝟐𝟐

MilewskiConstruction

Output perfect polyphase sequence

𝒔𝒔 = 𝑠𝑠 𝑛𝑛 𝑚𝑚=0𝒎𝒎𝟐𝟐𝟐𝟐+𝟏𝟏−1

perfect polyphase Sequence of length 𝑚𝑚

𝜷𝜷 = 𝛼𝛼 𝑛𝑛 𝑚𝑚=0𝑚𝑚−1

A positive integer

𝐾𝐾

where

𝒔𝒔 𝒏𝒏 = 𝜷𝜷 𝒒𝒒 𝝎𝝎𝒒𝒒𝒓𝒓

𝝎𝝎 = 𝒆𝒆−𝒋𝒋𝟐𝟐𝝅𝝅

𝒎𝒎𝟏𝟏+𝟐𝟐

Here, we use𝑛𝑛 = 𝒒𝒒𝒎𝒎𝟐𝟐 + 𝒓𝒓 ↔ (𝒒𝒒, 𝒓𝒓)

𝑚𝑚 � 𝑚𝑚𝐾𝐾 × 𝑚𝑚𝐾𝐾 array form of 𝒔𝒔

⋮ ⋮ ⋮⋱

⋮ ⋮ ⋮⋱

⋮ ⋮ ⋮⋱

𝛽𝛽(0) 𝛽𝛽(0) 𝛽𝛽(0)𝟏𝟏× × ×⋯ 𝟏𝟏𝟏𝟏𝛽𝛽(1) 𝛽𝛽(1) 𝛽𝛽(1)𝝎𝝎× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝟏𝟏𝟏𝟏

𝛽𝛽(2) 𝛽𝛽(2) 𝛽𝛽(2)𝝎𝝎𝟐𝟐× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝟐𝟐𝟏𝟏

𝛽𝛽(𝑚𝑚− 1) 𝛽𝛽(𝑚𝑚− 1) 𝛽𝛽(𝑚𝑚− 1)𝝎𝝎𝒎𝒎−𝟏𝟏× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝒎𝒎−𝟏𝟏𝟏𝟏

𝛽𝛽(0) 𝛽𝛽(0) 𝛽𝛽(0)𝝎𝝎𝒎𝒎(𝑵𝑵−𝟏𝟏)× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝒎𝒎(𝑵𝑵−𝟏𝟏)𝟏𝟏

𝛽𝛽(1) 𝛽𝛽(1) 𝛽𝛽(1)𝝎𝝎𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟏𝟏× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟏𝟏𝟏𝟏

𝛽𝛽(2) 𝛽𝛽(2) 𝛽𝛽(2)𝝎𝝎𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟐𝟐× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟐𝟐𝟏𝟏

𝛽𝛽(𝑚𝑚− 1) 𝛽𝛽(𝑚𝑚− 1) 𝛽𝛽(𝑚𝑚− 1)𝝎𝝎𝒎𝒎𝑵𝑵−𝟏𝟏× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝒎𝒎𝑵𝑵−𝟏𝟏𝟏𝟏

Here, 𝑁𝑁 = 𝑚𝑚𝐾𝐾 and the exponent runs from 0 to 𝑚𝑚𝑁𝑁 − 1

Input sequence 𝜷𝜷of period 𝑚𝑚

is repeated 𝑚𝑚𝐾𝐾 times

Hong-Yeop Song

The original Milewski constructionLength: 𝒎𝒎 → 𝒎𝒎 �𝒎𝒎𝟐𝟐𝟐𝟐

MilewskiConstruction

Output perfect polyphase sequence

𝒔𝒔 = 𝑠𝑠 𝑛𝑛 𝑚𝑚=0𝒎𝒎𝟐𝟐𝟐𝟐+𝟏𝟏−1

perfect polyphase Sequence of length 𝑚𝑚

𝜷𝜷 = 𝛼𝛼 𝑛𝑛 𝑚𝑚=0𝑚𝑚−1

A positive integer

𝐾𝐾

where

𝒔𝒔 𝒏𝒏 = 𝜷𝜷 𝒒𝒒 𝝎𝝎𝒒𝒒𝒓𝒓

𝝎𝝎 = 𝒆𝒆−𝒋𝒋𝟐𝟐𝝅𝝅

𝒎𝒎𝟏𝟏+𝟐𝟐

Here, we use𝑛𝑛 = 𝒒𝒒𝒎𝒎𝟐𝟐 + 𝒓𝒓 ↔ (𝒒𝒒, 𝒓𝒓)

𝑚𝑚 � 𝑚𝑚𝐾𝐾 × 𝑚𝑚𝐾𝐾 array form of 𝒔𝒔

⋮ ⋮ ⋮⋱

⋮ ⋮ ⋮⋱

⋮ ⋮ ⋮⋱

𝛽𝛽(0) 𝛽𝛽(0) 𝛽𝛽(0)𝟏𝟏× × ×⋯ 𝟏𝟏𝟏𝟏𝛽𝛽(1) 𝛽𝛽(1) 𝛽𝛽(1)𝝎𝝎× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝟏𝟏𝟏𝟏

𝛽𝛽(2) 𝛽𝛽(2) 𝛽𝛽(2)𝝎𝝎𝟐𝟐× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝟐𝟐𝟏𝟏

𝛽𝛽(𝑚𝑚− 1) 𝛽𝛽(𝑚𝑚− 1) 𝛽𝛽(𝑚𝑚− 1)𝝎𝝎𝒎𝒎−𝟏𝟏× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝒎𝒎−𝟏𝟏𝟏𝟏

𝛽𝛽(0) 𝛽𝛽(0) 𝛽𝛽(0)𝝎𝝎𝒎𝒎(𝑵𝑵−𝟏𝟏)× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝒎𝒎(𝑵𝑵−𝟏𝟏)𝟏𝟏

𝛽𝛽(1) 𝛽𝛽(1) 𝛽𝛽(1)𝝎𝝎𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟏𝟏× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟏𝟏𝟏𝟏

𝛽𝛽(2) 𝛽𝛽(2) 𝛽𝛽(2)𝝎𝝎𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟐𝟐× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟐𝟐𝟏𝟏

𝛽𝛽(𝑚𝑚− 1) 𝛽𝛽(𝑚𝑚− 1) 𝛽𝛽(𝑚𝑚− 1)𝝎𝝎𝒎𝒎𝑵𝑵−𝟏𝟏× × ×⋯ 𝝎𝝎𝑵𝑵−𝟏𝟏 𝒎𝒎𝑵𝑵−𝟏𝟏𝟏𝟏

Input sequence 𝜷𝜷of period 𝑚𝑚

is repeated 𝑚𝑚𝐾𝐾 times

the exponent runs from 0 to 𝑁𝑁 − 1

Hong-Yeop Song

Contents of Talk

• Some preliminary concepts

▫ Circular Florentine arrays

▫ Perfect sequences▫ Interleaved sequences▫ Milewski construction – review

• Proposed generalization

• 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 integer𝑁𝑁

A polyphase sequenceof length 𝑁𝑁

𝝁𝝁 𝜋𝜋

A functionℤ𝑵𝑵 ⟶ ℤ𝒎𝒎𝑵𝑵

A collection of 𝑁𝑁 sequence of length 𝑚𝑚𝑩𝑩 = 𝜷𝜷0,𝜷𝜷1, … ,𝜷𝜷𝑁𝑁

not necessarily polyphasenot necessarily all distinct

Hong-Yeop Song

Our framework(A special type of interleaved sequences)

where

with 𝒏𝒏 = 𝒒𝒒𝑵𝑵 + 𝒓𝒓, and 𝝎𝝎 = 𝐞𝐞𝐞𝐞𝐞𝐞(−𝒋𝒋𝟐𝟐𝝅𝝅/𝒎𝒎𝑵𝑵).

Interleaving technique

A positive integer𝑁𝑁

A polyphase sequenceof length 𝑁𝑁

𝝁𝝁 𝜋𝜋

A functionℤ𝑵𝑵 ⟶ ℤ𝒎𝒎𝑵𝑵

A collection of 𝑁𝑁 sequence of length 𝑚𝑚𝑩𝑩 = 𝜷𝜷0,𝜷𝜷1, … ,𝜷𝜷𝑁𝑁

not necessarily polyphasenot necessarily all distinct

Output sequence 𝑠𝑠 = 𝒔𝒔 𝒏𝒏 𝑚𝑚=0𝑚𝑚𝑁𝑁2−1

𝒔𝒔 𝒏𝒏 = 𝝁𝝁 𝒓𝒓 𝜷𝜷𝒓𝒓 𝒒𝒒 𝝎𝝎𝒒𝒒𝝅𝝅(𝒓𝒓)

Hong-Yeop Song

Our framework(A special type of interleaved sequences)

where

with 𝒏𝒏 = 𝒒𝒒𝑵𝑵 + 𝒓𝒓, and 𝝎𝝎 = 𝐞𝐞𝐞𝐞𝐞𝐞(−𝒋𝒋𝟐𝟐𝝅𝝅/𝒎𝒎𝑵𝑵).

Interleaving technique

A positive integer𝑁𝑁

A polyphase sequenceof length 𝑁𝑁

𝝁𝝁 𝜋𝜋

A functionℤ𝑵𝑵 ⟶ ℤ𝒎𝒎𝑵𝑵

A collection of 𝑁𝑁 sequence of length 𝑚𝑚𝑩𝑩 = 𝜷𝜷0,𝜷𝜷1, … ,𝜷𝜷𝑁𝑁

not necessarily polyphasenot necessarily all distinct

Output sequence 𝑠𝑠 = 𝒔𝒔 𝒏𝒏 𝑚𝑚=0𝑚𝑚𝑁𝑁2−1

𝒔𝒔 𝒏𝒏 = 𝝁𝝁 𝒓𝒓 𝜷𝜷𝒓𝒓 𝒒𝒒 𝝎𝝎𝒒𝒒𝝅𝝅(𝒓𝒓)

Definition. We define A 𝑩𝑩,𝝅𝝅 be a family of interleaved sequences constructed by the above procedure using all possible polyphase sequences 𝝁𝝁.

Hong-Yeop Song

Array FormAssume that 𝝁𝝁 is the all-one sequence, ※ 𝜔𝜔 = 𝑒𝑒−𝑗𝑗

2𝜋𝜋𝑚𝑚𝑁𝑁

Hong-Yeop Song

Array Form

Row index 𝒒𝒒

=𝟎𝟎,𝟏𝟏,𝟐𝟐,…

,𝒎𝒎𝑵𝑵−𝟏𝟏

Column index 𝒓𝒓 = 𝟎𝟎,𝟏𝟏,𝟐𝟐, … ,𝑵𝑵− 𝟏𝟏Assume that 𝝁𝝁 is the all-one sequence, ※ 𝜔𝜔 = 𝑒𝑒−𝑗𝑗

2𝜋𝜋𝑚𝑚𝑁𝑁

Hong-Yeop Song

Array Form

Row index 𝒒𝒒

=𝟎𝟎,𝟏𝟏,𝟐𝟐,…

,𝒎𝒎𝑵𝑵−𝟏𝟏

Column index 𝒓𝒓 = 𝟎𝟎,𝟏𝟏,𝟐𝟐, … ,𝑵𝑵− 𝟏𝟏Assume that 𝝁𝝁 is the all-one sequence, ※ 𝜔𝜔 = 𝑒𝑒−𝑗𝑗

2𝜋𝜋𝑚𝑚𝑁𝑁

⋮ ⋮ ⋮

⋮ ⋮ ⋮

⋮ ⋮ ⋮

𝛽𝛽0(0) 𝛽𝛽1(0) 𝛽𝛽𝑁𝑁−1(0)⋯

𝛽𝛽0(1) 𝛽𝛽1(1) 𝛽𝛽𝑁𝑁−1(1)⋯

𝛽𝛽0(2) 𝛽𝛽1(2) 𝛽𝛽𝑁𝑁−1(2)⋯

𝛽𝛽0(𝑚𝑚− 1) 𝛽𝛽1(𝑚𝑚− 1) 𝛽𝛽𝑁𝑁−1(𝑚𝑚− 1)⋯

𝛽𝛽0(0) 𝛽𝛽1(0) 𝛽𝛽𝑁𝑁−1(0)⋯

𝛽𝛽0(1) 𝛽𝛽1(1) 𝛽𝛽𝑁𝑁−1(1)⋯

𝛽𝛽0(2) 𝛽𝛽1(2) 𝛽𝛽𝑁𝑁−1(2)⋯

𝛽𝛽0(𝑚𝑚− 1) 𝛽𝛽1(𝑚𝑚− 1) 𝛽𝛽𝑁𝑁−1(𝑚𝑚− 1)⋯

Input sequence 𝜷𝜷0of period 𝑚𝑚

Repeating 𝑁𝑁 times

Input sequence 𝜷𝜷1of period 𝑚𝑚

Repeating 𝑁𝑁 times

Input sequence 𝜷𝜷𝑁𝑁−1of period 𝑚𝑚

Repeating 𝑁𝑁 times

Hong-Yeop Song

Array Form

Row index 𝒒𝒒

=𝟎𝟎,𝟏𝟏,𝟐𝟐,…

,𝒎𝒎𝑵𝑵−𝟏𝟏

Column index 𝒓𝒓 = 𝟎𝟎,𝟏𝟏,𝟐𝟐, … ,𝑵𝑵− 𝟏𝟏Assume that 𝝁𝝁 is the all-one sequence, ※ 𝜔𝜔 = 𝑒𝑒−𝑗𝑗

2𝜋𝜋𝑚𝑚𝑁𝑁

⋮ ⋮ ⋮⋱

⋮ ⋮ ⋮⋱

⋮ ⋮ ⋮⋱

𝛽𝛽0(0) 𝛽𝛽1(0) 𝛽𝛽𝑁𝑁−1(0)𝜔𝜔𝜋𝜋(1) 𝟎𝟎× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝟎𝟎𝜔𝜔𝜋𝜋(0) 0

𝛽𝛽0(1) 𝛽𝛽1(1) 𝛽𝛽𝑁𝑁−1(1)𝜔𝜔𝜋𝜋(1) 𝟏𝟏× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝟏𝟏𝜔𝜔𝜋𝜋(0) 𝟏𝟏

𝛽𝛽0(2) 𝛽𝛽1(2) 𝛽𝛽𝑁𝑁−1(2)𝜔𝜔𝜋𝜋(1) 𝟐𝟐× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝟐𝟐𝜔𝜔𝜋𝜋(0) 𝟐𝟐

𝛽𝛽0(𝑚𝑚− 1) 𝛽𝛽1(𝑚𝑚− 1) 𝛽𝛽𝑁𝑁−1(𝑚𝑚− 1)𝜔𝜔𝜋𝜋(1) 𝒎𝒎−𝟏𝟏× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝒎𝒎−𝟏𝟏𝜔𝜔𝜋𝜋(0) (𝒎𝒎−𝟏𝟏)

𝛽𝛽0(0) 𝛽𝛽1(0) 𝛽𝛽𝑁𝑁−1(0)𝜔𝜔𝜋𝜋(1) 𝒎𝒎(𝑵𝑵−𝟏𝟏)× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝒎𝒎(𝑵𝑵−𝟏𝟏)𝜔𝜔𝜋𝜋(0) 𝒎𝒎(𝑵𝑵−𝟏𝟏)

𝛽𝛽0(1) 𝛽𝛽1(1) 𝛽𝛽𝑁𝑁−1(1)𝜔𝜔𝜋𝜋(1) 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟏𝟏× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟏𝟏𝜔𝜔𝜋𝜋(0) 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟏𝟏

𝛽𝛽0(2) 𝛽𝛽1(2) 𝛽𝛽𝑁𝑁−1(2)𝜔𝜔𝜋𝜋(1) 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟐𝟐× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟐𝟐𝜔𝜔𝜋𝜋(0) 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟐𝟐

𝛽𝛽0(𝑚𝑚− 1) 𝛽𝛽1(𝑚𝑚− 1) 𝛽𝛽𝑁𝑁−1(𝑚𝑚− 1)𝜔𝜔𝜋𝜋(1) 𝒎𝒎𝑵𝑵−𝟏𝟏× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝒎𝒎𝑵𝑵−𝟏𝟏𝜔𝜔𝜋𝜋(0) 𝒎𝒎𝑵𝑵−𝟏𝟏

Input function 𝜋𝜋:ℤ𝑵𝑵 ⟶ ℤ𝒎𝒎𝑵𝑵

Hong-Yeop Song

Array Form

Row index 𝒒𝒒

=𝟎𝟎,𝟏𝟏,𝟐𝟐,…

,𝒎𝒎𝑵𝑵−𝟏𝟏

Column index 𝒓𝒓 = 𝟎𝟎,𝟏𝟏,𝟐𝟐, … ,𝑵𝑵− 𝟏𝟏Assume that 𝝁𝝁 is the all-one sequence,

⋮ ⋮ ⋮⋱

⋮ ⋮ ⋮⋱

⋮ ⋮ ⋮⋱

𝛽𝛽0(0) 𝛽𝛽1(0) 𝛽𝛽𝑁𝑁−1(0)𝜔𝜔𝜋𝜋(1) 𝟎𝟎× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝟎𝟎𝜔𝜔𝜋𝜋(0) 0

𝛽𝛽0(1) 𝛽𝛽1(1) 𝛽𝛽𝑁𝑁−1(1)𝜔𝜔𝜋𝜋(1) 𝟏𝟏× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝟏𝟏𝜔𝜔𝜋𝜋(0) 𝟏𝟏

𝛽𝛽0(2) 𝛽𝛽1(2) 𝛽𝛽𝑁𝑁−1(2)𝜔𝜔𝜋𝜋(1) 𝟐𝟐× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝟐𝟐𝜔𝜔𝜋𝜋(0) 𝟐𝟐

𝛽𝛽0(𝑚𝑚− 1) 𝛽𝛽1(𝑚𝑚− 1) 𝛽𝛽𝑁𝑁−1(𝑚𝑚− 1)𝜔𝜔𝜋𝜋(1) 𝒎𝒎−𝟏𝟏× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝒎𝒎−𝟏𝟏𝜔𝜔𝜋𝜋(0) (𝒎𝒎−𝟏𝟏)

𝛽𝛽0(0) 𝛽𝛽1(0) 𝛽𝛽𝑁𝑁−1(0)𝜔𝜔𝜋𝜋(1) 𝒎𝒎(𝑵𝑵−𝟏𝟏)× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝒎𝒎(𝑵𝑵−𝟏𝟏)𝜔𝜔𝜋𝜋(0) 𝒎𝒎(𝑵𝑵−𝟏𝟏)

𝛽𝛽0(1) 𝛽𝛽1(1) 𝛽𝛽𝑁𝑁−1(1)𝜔𝜔𝜋𝜋(1) 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟏𝟏× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟏𝟏𝜔𝜔𝜋𝜋(0) 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟏𝟏

𝛽𝛽0(2) 𝛽𝛽1(2) 𝛽𝛽𝑁𝑁−1(2)𝜔𝜔𝜋𝜋(1) 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟐𝟐× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟐𝟐𝜔𝜔𝜋𝜋(0) 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟐𝟐

𝛽𝛽0(𝑚𝑚− 1) 𝛽𝛽1(𝑚𝑚− 1) 𝛽𝛽𝑁𝑁−1(𝑚𝑚− 1)𝜔𝜔𝜋𝜋(1) 𝒎𝒎𝑵𝑵−𝟏𝟏× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝒎𝒎𝑵𝑵−𝟏𝟏𝜔𝜔𝜋𝜋(0) 𝒎𝒎𝑵𝑵−𝟏𝟏

the exponent runs from 𝜋𝜋(0) to 𝜋𝜋(𝑁𝑁 − 1)

Hong-Yeop Song

Array Form

Row index 𝒒𝒒

=𝟎𝟎,𝟏𝟏,𝟐𝟐,…

,𝒎𝒎𝑵𝑵−𝟏𝟏

Column index 𝒓𝒓 = 𝟎𝟎,𝟏𝟏,𝟐𝟐, … ,𝑵𝑵− 𝟏𝟏Assume that 𝝁𝝁 is the all-one sequence, ※ 𝜔𝜔 = 𝑒𝑒−𝑗𝑗

2𝜋𝜋𝑚𝑚𝑁𝑁

⋮ ⋮ ⋮⋱

⋮ ⋮ ⋮⋱

⋮ ⋮ ⋮⋱

𝛽𝛽0(0) 𝛽𝛽1(0) 𝛽𝛽𝑁𝑁−1(0)𝜔𝜔𝜋𝜋(1) 𝟎𝟎× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝟎𝟎𝜔𝜔𝜋𝜋(0) 0

𝛽𝛽0(1) 𝛽𝛽1(1) 𝛽𝛽𝑁𝑁−1(1)𝜔𝜔𝜋𝜋(1) 𝟏𝟏× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝟏𝟏𝜔𝜔𝜋𝜋(0) 𝟏𝟏

𝛽𝛽0(2) 𝛽𝛽1(2) 𝛽𝛽𝑁𝑁−1(2)𝜔𝜔𝜋𝜋(1) 𝟐𝟐× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝟐𝟐𝜔𝜔𝜋𝜋(0) 𝟐𝟐

𝛽𝛽0(𝑚𝑚− 1) 𝛽𝛽1(𝑚𝑚− 1) 𝛽𝛽𝑁𝑁−1(𝑚𝑚− 1)𝜔𝜔𝜋𝜋(1) 𝒎𝒎−𝟏𝟏× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝒎𝒎−𝟏𝟏𝜔𝜔𝜋𝜋(0) (𝒎𝒎−𝟏𝟏)

𝛽𝛽0(0) 𝛽𝛽1(0) 𝛽𝛽𝑁𝑁−1(0)𝜔𝜔𝜋𝜋(1) 𝒎𝒎(𝑵𝑵−𝟏𝟏)× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝒎𝒎(𝑵𝑵−𝟏𝟏)𝜔𝜔𝜋𝜋(0) 𝒎𝒎(𝑵𝑵−𝟏𝟏)

𝛽𝛽0(1) 𝛽𝛽1(1) 𝛽𝛽𝑁𝑁−1(1)𝜔𝜔𝜋𝜋(1) 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟏𝟏× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟏𝟏𝜔𝜔𝜋𝜋(0) 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟏𝟏

𝛽𝛽0(2) 𝛽𝛽1(2) 𝛽𝛽𝑁𝑁−1(2)𝜔𝜔𝜋𝜋(1) 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟐𝟐× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟐𝟐𝜔𝜔𝜋𝜋(0) 𝒎𝒎 𝑵𝑵−𝟏𝟏 +𝟐𝟐

𝛽𝛽0(𝑚𝑚− 1) 𝛽𝛽1(𝑚𝑚− 1) 𝛽𝛽𝑁𝑁−1(𝑚𝑚− 1)𝜔𝜔𝜋𝜋(1) 𝒎𝒎𝑵𝑵−𝟏𝟏× × ×⋯ 𝜔𝜔𝜋𝜋(𝑁𝑁−1) 𝒎𝒎𝑵𝑵−𝟏𝟏𝜔𝜔𝜋𝜋(0) 𝒎𝒎𝑵𝑵−𝟏𝟏

Here, the exponent of 𝝎𝝎𝝅𝝅(𝒓𝒓)

runs from 𝟎𝟎 to 𝒎𝒎𝑵𝑵− 𝟏𝟏

Hong-Yeop Song

Milewski Construction is a Special Case

MilewskiConstruction

Output perfect polyphase sequence

𝒔𝒔 = 𝑠𝑠 𝑛𝑛 𝑚𝑚=0𝒎𝒎𝟐𝟐𝟐𝟐+𝟏𝟏−1

perfect polyphase sequence

𝜷𝜷 = 𝛼𝛼 𝑛𝑛 𝑚𝑚=0𝑚𝑚−1

A positive integer

𝐾𝐾

where𝑠𝑠 𝑛𝑛 = 𝛽𝛽 𝑞𝑞 𝜔𝜔𝑞𝑞𝒓𝒓

Interleaving technique

𝑁𝑁 𝝁𝝁 𝜋𝜋𝐵𝐵 = 𝜷𝜷0,𝜷𝜷1, … ,𝜷𝜷𝑁𝑁

Output sequence

=?

Hong-Yeop Song

Milewski Construction is a Special Case

MilewskiConstruction

Output perfect polyphase sequence

𝒔𝒔 = 𝑠𝑠 𝑛𝑛 𝑚𝑚=0𝒎𝒎𝟐𝟐𝟐𝟐+𝟏𝟏−1

perfect polyphase sequence

𝜷𝜷 = 𝛼𝛼 𝑛𝑛 𝑚𝑚=0𝑚𝑚−1

A positive integer

𝐾𝐾

where𝑠𝑠 𝑛𝑛 = 𝛽𝛽 𝑞𝑞 𝜔𝜔𝑞𝑞𝒓𝒓

Interleaving technique

𝑁𝑁 𝝁𝝁 𝜋𝜋𝐵𝐵 = 𝜷𝜷0,𝜷𝜷1, … ,𝜷𝜷𝑁𝑁

Output sequence

𝒔𝒔 = 𝑠𝑠 𝑛𝑛 𝑚𝑚=0𝒎𝒎 𝒎𝒎𝟐𝟐 𝟐𝟐

−1

perfect polyphase sequences

𝜷𝜷0 = 𝜷𝜷1 = ⋯ = 𝜷𝜷𝑁𝑁−1An integer𝑁𝑁 = 𝑚𝑚𝐾𝐾

all-one sequence

The identityfunction𝜋𝜋 𝑟𝑟 = 𝑟𝑟

=

Hong-Yeop Song

Milewski Construction is a Special Case

MilewskiConstruction

Output perfect polyphase sequence

𝒔𝒔 = 𝑠𝑠 𝑛𝑛 𝑚𝑚=0𝒎𝒎𝟐𝟐𝟐𝟐+𝟏𝟏−1

perfect polyphase sequence

𝜷𝜷 = 𝛼𝛼 𝑛𝑛 𝑚𝑚=0𝑚𝑚−1

A positive integer

𝐾𝐾

where𝑠𝑠 𝑛𝑛 = 𝜷𝜷 𝒒𝒒 𝝎𝝎𝒒𝒒𝒓𝒓

Interleaving technique

𝑁𝑁 𝝁𝝁

where𝑠𝑠 𝑛𝑛 = 𝛽𝛽𝒓𝒓 𝑞𝑞 𝜔𝜔𝑞𝑞𝝅𝝅(𝒓𝒓) = 𝜷𝜷 𝒒𝒒 𝝎𝝎𝒒𝒒𝒓𝒓

𝜋𝜋𝐵𝐵 = 𝜷𝜷0,𝜷𝜷1, … ,𝜷𝜷𝑁𝑁

Output sequence

𝒔𝒔 = 𝑠𝑠 𝑛𝑛 𝑚𝑚=0𝒎𝒎 𝒎𝒎𝟐𝟐 𝟐𝟐

−1

perfect polyphase sequences

𝜷𝜷0 = 𝜷𝜷1 = ⋯ = 𝜷𝜷𝑁𝑁−1An integer𝑁𝑁 = 𝑚𝑚𝐾𝐾

all-one sequence

The identityfunction𝜋𝜋 𝑟𝑟 = 𝑟𝑟

=

Hong-Yeop Song

Important Link

Definition. Let 𝝅𝝅,𝝈𝝈 be two functions from ℤ𝑵𝑵 to ℤ𝒎𝒎𝑵𝑵. We define

When 𝝅𝝅 = 𝝈𝝈, we use 𝜳𝜳𝝅𝝅 𝝉𝝉 simply.

𝜳𝜳𝝅𝝅,𝝈𝝈 𝝉𝝉 = 𝒙𝒙 ∈ ℤ𝑵𝑵 𝝅𝝅 𝒙𝒙 + 𝝉𝝉 ≡ 𝝈𝝈 𝒙𝒙 𝒎𝒎𝒎𝒎𝒎𝒎 𝑵𝑵 .

Hong-Yeop Song

Condition on perfectness(Main result 1)

Theorem. Any sequence inA 𝑩𝑩,𝝅𝝅 is perfect if and only if the following conditions are satisfied:

1) 𝚿𝚿𝝅𝝅(𝒓𝒓) = 𝟎𝟎 for 𝒓𝒓 = 𝟏𝟏,𝟐𝟐, … ,𝑵𝑵− 𝟏𝟏.That is, 𝝅𝝅 𝒓𝒓 (𝐦𝐦𝐦𝐦𝐦𝐦 𝑵𝑵) for 𝒓𝒓 = 𝟎𝟎,𝟏𝟏, … ,𝑵𝑵− 𝟏𝟏 is a permutation over ℤ𝑵𝑵.

2) 𝑩𝑩 is a collection of perfect sequences all of period 𝑚𝑚 with the same energy.

Definition. Let 𝝅𝝅,𝝈𝝈 be two functions from ℤ𝑵𝑵 to ℤ𝒎𝒎𝑵𝑵. We define

When 𝝅𝝅 = 𝝈𝝈, we use 𝜳𝜳𝝅𝝅 𝝉𝝉 simply.

𝜳𝜳𝝅𝝅,𝝈𝝈 𝝉𝝉 = 𝒙𝒙 ∈ ℤ𝑵𝑵 𝝅𝝅 𝒙𝒙 + 𝝉𝝉 ≡ 𝝈𝝈 𝒙𝒙 𝒎𝒎𝒎𝒎𝒎𝒎 𝑵𝑵 .

Hong-Yeop Song

Condition on perfectness(Main result 1)

Theorem. Any sequence inA 𝑩𝑩,𝝅𝝅 is perfect if and only if the following conditions are satisfied:

1) 𝚿𝚿𝝅𝝅(𝒓𝒓) = 𝟎𝟎 for 𝒓𝒓 = 𝟏𝟏,𝟐𝟐, … ,𝑵𝑵− 𝟏𝟏.That is, 𝝅𝝅 𝒓𝒓 (𝐦𝐦𝐦𝐦𝐦𝐦 𝑵𝑵) for 𝒓𝒓 = 𝟎𝟎,𝟏𝟏, … ,𝑵𝑵− 𝟏𝟏 is a permutation over ℤ𝑵𝑵.

2) 𝑩𝑩 is a collection of perfect sequences all of period 𝑚𝑚 with the same energy.

We now call themthe generalized Milewski sequences

Definition. Let 𝝅𝝅,𝝈𝝈 be two functions from ℤ𝑵𝑵 to ℤ𝒎𝒎𝑵𝑵. We define

When 𝝅𝝅 = 𝝈𝝈, we use 𝜳𝜳𝝅𝝅 𝝉𝝉 simply.

𝜳𝜳𝝅𝝅,𝝈𝝈 𝝉𝝉 = 𝒙𝒙 ∈ ℤ𝑵𝑵 𝝅𝝅 𝒙𝒙 + 𝝉𝝉 ≡ 𝝈𝝈 𝒙𝒙 𝒎𝒎𝒎𝒎𝒎𝒎 𝑵𝑵 .

Hong-Yeop Song

Examples

Generalized Milewski

Construction

𝑁𝑁 𝝁𝝁𝐵𝐵 = 𝜷𝜷0,𝜷𝜷1 𝜋𝜋

• 𝜷𝜷0 = 𝜷𝜷1 = 0,−1,1,0,1,1which is a perfect sequence of length 6,

• 𝑁𝑁 = 2,• 𝜋𝜋(𝑟𝑟) = 𝑟𝑟, and• 𝝁𝝁 is the all-one sequence.

Hong-Yeop Song

Examples

※ 𝜔𝜔 = 𝑒𝑒−𝑗𝑗2𝜋𝜋12

Generalized Milewski

Construction

𝑁𝑁 𝝁𝝁𝐵𝐵 = 𝜷𝜷0,𝜷𝜷1 𝜋𝜋

𝒔𝒔 = 0, 0,−1,−𝜔𝜔, 1,𝜔𝜔2, 0, 0, 1,𝜔𝜔4, 1,𝜔𝜔5, 0, 0,−1,−𝜔𝜔7, 1,𝜔𝜔8, 0, 0, 1,𝜔𝜔10, 1,𝜔𝜔11

is a perfect sequence of length 24.

• 𝜷𝜷0 = 𝜷𝜷1 = 0,−1,1,0,1,1which is a perfect sequence of length 6,

• 𝑁𝑁 = 2,• 𝜋𝜋(𝑟𝑟) = 𝑟𝑟, and• 𝝁𝝁 is the all-one sequence.

Hong-Yeop Song

ExamplesConstellation of

𝒔𝒔

※ 𝜔𝜔 = 𝑒𝑒−𝑗𝑗2𝜋𝜋12

Generalized Milewski

Construction

𝑁𝑁 𝝁𝝁𝐵𝐵 = 𝜷𝜷0,𝜷𝜷1 𝜋𝜋

𝒔𝒔 = 0, 0,−1,−𝜔𝜔, 1,𝜔𝜔2, 0, 0, 1,𝜔𝜔4, 1,𝜔𝜔5, 0, 0,−1,−𝜔𝜔7, 1,𝜔𝜔8, 0, 0, 1,𝜔𝜔10, 1,𝜔𝜔11

is a perfect sequence of length 24.

• 𝜷𝜷0 = 𝜷𝜷1 = 0,−1,1,0,1,1which is a perfect sequence of length 6,

• 𝑁𝑁 = 2,• 𝜋𝜋(𝑟𝑟) = 𝑟𝑟, and• 𝝁𝝁 is the all-one sequence.

Hong-Yeop Song

Examples

※ 𝜔𝜔 = 𝑒𝑒− 𝑗𝑗2𝜋𝜋12

Generalized Milewski

Construction

𝑁𝑁 𝝁𝝁𝐵𝐵 = 𝜷𝜷0,𝜷𝜷1,𝜷𝜷2 𝜋𝜋

𝒔𝒔 is a perfect sequence of length 90.

• 𝜷𝜷0 = 𝜷𝜷1 = 𝜷𝜷2 =3,−2,3,−2,−2,3,−2,−7,−2,−2

which is a perfect sequence of period 10• 𝑁𝑁 = 3,• 𝜋𝜋(𝑟𝑟) = 𝑟𝑟, and• 𝝁𝝁 is the all-one sequence.

ASK constellation

Hong-Yeop Song

Examples

※ 𝜔𝜔 = 𝑒𝑒− 𝑗𝑗2𝜋𝜋12

Generalized Milewski

Construction

𝑁𝑁 𝝁𝝁𝐵𝐵 = 𝜷𝜷0,𝜷𝜷1,𝜷𝜷2 𝜋𝜋

𝒔𝒔 is a perfect sequence of length 90.

• 𝜷𝜷0 = 𝜷𝜷1 = 𝜷𝜷2 =3,−2,3,−2,−2,3,−2,−7,−2,−2

which is a perfect sequence of period 10• 𝑁𝑁 = 3,• 𝜋𝜋(𝑟𝑟) = 𝑟𝑟, and• 𝝁𝝁 is the all-one sequence.

Constellation of 𝒔𝒔

APSK constellation

ASK constellation

Hong-Yeop Song

Direct vs Indirect (when 𝑁𝑁 is composite)

Perfect sequences of length 𝑚𝑚

Generalized Milewski

sequences of length 𝑚𝑚𝑁𝑁12

Generalized Milewski

sequences of length 𝑚𝑚𝑁𝑁2

Two-step synthesis

Direct synthesis

Hong-Yeop Song

Theorem. Assume that 𝑁𝑁 is a composite number.1) Any generalized Milewski sequence of length 𝑚𝑚𝑁𝑁2 from the two-step

method can be also obtained by the direct method.2) There exists a generalized Milewski sequence of length 𝑚𝑚𝑁𝑁2 from the direct

method which can not be obtained by the two-step method.

Perfect sequences of length 𝑚𝑚

Generalized Milewski

sequences of length 𝑚𝑚𝑁𝑁12

Generalized Milewski

sequences of length 𝑚𝑚𝑁𝑁2

Two-step synthesis

Direct synthesis

Direct vs Indirect (when 𝑁𝑁 is composite)

Hong-Yeop Song

Condition on optimal pair(Main result 2)

Theorem. Let 𝐵𝐵1 = 𝜷𝜷0,𝜷𝜷1, … ,𝜷𝜷𝑁𝑁−1 and 𝐵𝐵2 = 𝜸𝜸0,𝜸𝜸1, … ,𝜸𝜸𝑁𝑁−1 , all of length 𝑚𝑚 and the same energy 𝐸𝐸𝐵𝐵, and perfect.

Construct 𝒔𝒔 ∈ A 𝐵𝐵1,𝜋𝜋 and 𝒇𝒇 ∈ A(𝐵𝐵2,𝜎𝜎).

Hong-Yeop Song

Condition on optimal pair(Main result 2)

Theorem. Let 𝐵𝐵1 = 𝜷𝜷0,𝜷𝜷1, … ,𝜷𝜷𝑁𝑁−1 and 𝐵𝐵2 = 𝜸𝜸0,𝜸𝜸1, … ,𝜸𝜸𝑁𝑁−1 , all of length 𝑚𝑚 and the same energy 𝐸𝐸𝐵𝐵, and perfect.

Construct 𝒔𝒔 ∈ A 𝐵𝐵1,𝜋𝜋 and 𝒇𝒇 ∈ A(𝐵𝐵2,𝜎𝜎).

Then, 𝒔𝒔 and 𝒇𝒇 have optimal crosscorrelation if and only if the following conditions are satisfied for each 𝒓𝒓 = 𝟎𝟎,𝟏𝟏, … ,𝑵𝑵− 𝟏𝟏:1) Ψ𝜋𝜋,𝜎𝜎(𝑟𝑟) = 1, i.e., Ψ𝜋𝜋,𝜎𝜎 𝑟𝑟 = 𝑥𝑥 .2) For the unique 𝑥𝑥 ∈ Ψ𝜋𝜋,𝜎𝜎(𝑟𝑟), the pair of sequences

𝛽𝛽𝑥𝑥+𝑟𝑟 𝑡𝑡 𝜔𝜔𝑚𝑚𝜋𝜋 𝑥𝑥+𝑟𝑟 𝑡𝑡

𝑡𝑡=0

𝑚𝑚−1and 𝛾𝛾𝑥𝑥 𝑡𝑡 𝜔𝜔𝑚𝑚

𝜎𝜎 𝑥𝑥 𝑡𝑡𝑡𝑡=0

𝑚𝑚−1is optimal.

Hong-Yeop Song

Condition on optimal pair(Simple Special Case)

Corollary. Let 𝐵𝐵1 = 𝜷𝜷0,𝜷𝜷1, … ,𝜷𝜷𝑁𝑁−1 and 𝐵𝐵2 = 𝜸𝜸0,𝜸𝜸1, … ,𝜸𝜸𝑁𝑁−1 , all of length 𝑚𝑚 and the same energy 𝐸𝐸𝐵𝐵, and perfect.

Assume that 𝝅𝝅 and 𝝈𝝈 have the same range.Construct 𝒔𝒔 ∈ A 𝐵𝐵1,𝜋𝜋 and 𝒇𝒇 ∈ A(𝐵𝐵2,𝜎𝜎).

Hong-Yeop Song

Condition on optimal pair(Simple Special Case)

Then, 𝒔𝒔 and 𝒇𝒇 have optimal crosscorrelation if and only if the following conditions are satisfied for each 𝒓𝒓 = 𝟎𝟎,𝟏𝟏, … ,𝑵𝑵− 𝟏𝟏:1) Ψ𝜋𝜋,𝜎𝜎(𝑟𝑟) = 1, i.e., Ψ𝜋𝜋,𝜎𝜎 𝑟𝑟 = 𝑥𝑥 .2) For the unique 𝑥𝑥 ∈ Ψ𝜋𝜋,𝜎𝜎(𝑟𝑟),

the pair of sequences 𝜷𝜷𝒙𝒙+𝒓𝒓 and 𝜸𝜸𝒙𝒙 is optimal.

Corollary. Let 𝐵𝐵1 = 𝜷𝜷0,𝜷𝜷1, … ,𝜷𝜷𝑁𝑁−1 and 𝐵𝐵2 = 𝜸𝜸0,𝜸𝜸1, … ,𝜸𝜸𝑁𝑁−1 , all of length 𝑚𝑚 and the same energy 𝐸𝐸𝐵𝐵, and perfect.

Assume that 𝝅𝝅 and 𝝈𝝈 have the same range.Construct 𝒔𝒔 ∈ A 𝐵𝐵1,𝜋𝜋 and 𝒇𝒇 ∈ A(𝐵𝐵2,𝜎𝜎).

Hong-Yeop Song

Condition on optimal pair(Simple Special Case)

Then, 𝒔𝒔 and 𝒇𝒇 have optimal crosscorrelation if and only if the following conditions are satisfied for each 𝒓𝒓 = 𝟎𝟎,𝟏𝟏, … ,𝑵𝑵− 𝟏𝟏:1) Ψ𝜋𝜋,𝜎𝜎(𝑟𝑟) = 1, i.e., Ψ𝜋𝜋,𝜎𝜎 𝑟𝑟 = 𝑥𝑥 .2) For the unique 𝑥𝑥 ∈ Ψ𝜋𝜋,𝜎𝜎(𝑟𝑟),

the pair of sequences 𝜷𝜷𝒙𝒙+𝒓𝒓 and 𝜸𝜸𝒙𝒙 is optimal.

Corollary. Let 𝐵𝐵1 = 𝜷𝜷0,𝜷𝜷1, … ,𝜷𝜷𝑁𝑁−1 and 𝐵𝐵2 = 𝜸𝜸0,𝜸𝜸1, … ,𝜸𝜸𝑁𝑁−1 , all of length 𝑚𝑚 and the same energy 𝐸𝐸𝐵𝐵, and perfect.

Assume that 𝝅𝝅 and 𝝈𝝈 have the same range.Construct 𝒔𝒔 ∈ A 𝐵𝐵1,𝜋𝜋 and 𝒇𝒇 ∈ A(𝐵𝐵2,𝜎𝜎).

Condition on input pairs

Hong-Yeop Song

Condition on optimal pair(Simple Special Case)

Then, 𝒔𝒔 and 𝒇𝒇 have optimal crosscorrelation if and only if the following conditions are satisfied for each 𝒓𝒓 = 𝟎𝟎,𝟏𝟏, … ,𝑵𝑵− 𝟏𝟏:1) Ψ𝜋𝜋,𝜎𝜎(𝑟𝑟) = 1, i.e., Ψ𝜋𝜋,𝜎𝜎 𝑟𝑟 = 𝑥𝑥 .2) For the unique 𝑥𝑥 ∈ Ψ𝜋𝜋,𝜎𝜎(𝑟𝑟),

the pair of sequences 𝜷𝜷𝒙𝒙+𝒓𝒓 and 𝜸𝜸𝒙𝒙 is optimal.

Corollary. Let 𝐵𝐵1 = 𝜷𝜷0,𝜷𝜷1, … ,𝜷𝜷𝑁𝑁−1 and 𝐵𝐵2 = 𝜸𝜸0,𝜸𝜸1, … ,𝜸𝜸𝑁𝑁−1 , all of length 𝑚𝑚 and the same energy 𝐸𝐸𝐵𝐵, and perfect.

Assume that 𝝅𝝅 and 𝝈𝝈 have the same range.Construct 𝒔𝒔 ∈ A 𝐵𝐵1,𝜋𝜋 and 𝒇𝒇 ∈ A(𝐵𝐵2,𝜎𝜎).

Condition on input pairs

Condition on input

permutations

Hong-Yeop Song

when 𝒎𝒎 = 𝟏𝟏• The all-one sequence of length 1 is a trivial perfect sequence.• Therefore,

“the all-one sequence and itself is a (trivial) optimal pair of perfect sequences of length 1”

Hong-Yeop Song

when 𝒎𝒎 = 𝟏𝟏• The all-one sequence of length 1 is a trivial perfect sequence.• Therefore,

“the all-one sequence and itself is a (trivial) optimal pair of perfect sequences of length 1”

an optimal 𝑘𝑘-set of generalized Milewski sequences of length 𝑵𝑵𝟐𝟐 exists

if and only if a 𝑘𝑘 × 𝑁𝑁 circular Florentine array exists

• Therefore, for 𝑚𝑚 = 1, we have:

Hong-Yeop Song

Example

• For a 4 × 15 circular Florentine array

we have optimal 𝟒𝟒-set of generalized Milewski sequences of length 𝑵𝑵𝟐𝟐 = 𝟏𝟏𝟏𝟏𝟐𝟐 by picking up any single perfect sequence from each and every family

𝜋𝜋1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

𝜋𝜋2 0 7 1 8 2 12 3 11 9 4 13 5 14 6 10

𝜋𝜋3 0 4 11 7 10 1 13 9 5 8 3 6 2 14 12

𝜋𝜋4 0 13 7 2 11 6 14 10 3 5 12 9 1 4 8

A 1 ,𝜋𝜋1 , A 1 ,𝜋𝜋2 , A 1 ,𝜋𝜋3 , and

A 1 ,𝜋𝜋4 .

(Song, 91 and 00)

Hong-Yeop Song

Example

• For a 𝟒𝟒 × 𝟏𝟏𝟏𝟏 circular Florentine array

we have optimal 𝟒𝟒-set of generalized Milewski sequences of length 𝑵𝑵𝟐𝟐 = 𝟏𝟏𝟏𝟏𝟐𝟐 by picking up any single perfect sequence from each and every family

• New, in the sense of size 4 (previously known only of size 2) for length 𝟏𝟏𝟏𝟏𝟐𝟐 or 𝟏𝟏𝟏𝟏

𝜋𝜋1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

𝜋𝜋2 0 7 1 8 2 12 3 11 9 4 13 5 14 6 10

𝜋𝜋3 0 4 11 7 10 1 13 9 5 8 3 6 2 14 12

𝜋𝜋4 0 13 7 2 11 6 14 10 3 5 12 9 1 4 8

A 1 ,𝜋𝜋1 , A 1 ,𝜋𝜋2 , A 1 ,𝜋𝜋3 , and

A 1 ,𝜋𝜋4

(Song, 91 and 00)

Hong-Yeop Song

when 𝒎𝒎 > 𝟏𝟏

𝝅𝝅𝟏𝟏 0 1 2 3 4

𝝅𝝅𝟐𝟐 0 2 4 1 3Assume we have an optimal pair 𝜷𝜷,𝜸𝜸and a 𝟐𝟐 × 𝟏𝟏 circular Florentine array:

Hong-Yeop Song

when 𝒎𝒎 > 𝟏𝟏

𝝅𝝅𝟏𝟏 0 1 2 3 4

𝝅𝝅𝟐𝟐 0 2 4 1 3

1) Ψ𝜋𝜋,𝜎𝜎(𝑟𝑟) = 1, i.e., Ψ𝜋𝜋,𝜎𝜎 𝑟𝑟 = 𝑥𝑥 .2) For the unique 𝑥𝑥 ∈ Ψ𝜋𝜋,𝜎𝜎(𝑟𝑟), the pair of sequences 𝜷𝜷𝒙𝒙+𝒓𝒓 and 𝜸𝜸𝒙𝒙 is optimal.

Assume we have an optimal pair 𝜷𝜷,𝜸𝜸and a 𝟐𝟐 × 𝟏𝟏 circular Florentine array:

Hong-Yeop Song

when 𝒎𝒎 > 𝟏𝟏

• Construct 𝑠𝑠 ∈ A 𝐵𝐵1,𝜋𝜋1 , 𝑓𝑓 ∈ A 𝐵𝐵2,𝜋𝜋2 with𝐵𝐵1 = 𝛽𝛽0,𝛽𝛽1, … ,𝛽𝛽𝑁𝑁−1 and 𝐵𝐵2 = 𝛾𝛾0, 𝛾𝛾1, … , 𝛾𝛾𝑁𝑁−1 , where

𝝅𝝅𝟏𝟏 0 1 2 3 4

𝝅𝝅𝟐𝟐 0 2 4 1 3

𝜷𝜷0𝜷𝜷1

𝜷𝜷4

⋮

= 𝜷𝜷= 𝜷𝜷

= 𝜷𝜷

𝜸𝜸0𝜸𝜸1

𝜸𝜸4

⋮

= 𝜸𝜸= 𝜸𝜸

= 𝜸𝜸

1) Ψ𝜋𝜋,𝜎𝜎(𝑟𝑟) = 1, i.e., Ψ𝜋𝜋,𝜎𝜎 𝑟𝑟 = 𝑥𝑥 .2) For the unique 𝑥𝑥 ∈ Ψ𝜋𝜋,𝜎𝜎(𝑟𝑟), the pair of sequences 𝜷𝜷𝒙𝒙+𝒓𝒓 and 𝜸𝜸𝒙𝒙 is optimal.

Assume we have an optimal pair 𝜷𝜷,𝜸𝜸and a 𝟐𝟐 × 𝟏𝟏 circular Florentine array:

Hong-Yeop Song

when 𝒎𝒎 > 𝟏𝟏

• Construct 𝑠𝑠 ∈ A 𝐵𝐵1,𝜋𝜋1 , 𝑓𝑓 ∈ A 𝐵𝐵2,𝜋𝜋2 with𝐵𝐵1 = 𝛽𝛽0,𝛽𝛽1, … ,𝛽𝛽𝑁𝑁−1 and 𝐵𝐵2 = 𝛾𝛾0, 𝛾𝛾1, … , 𝛾𝛾𝑁𝑁−1 , where

𝝅𝝅𝟏𝟏 0 1 2 3 4

𝝅𝝅𝟐𝟐 0 2 4 1 3

𝜷𝜷0𝜷𝜷1

𝜷𝜷4

⋮

= 𝜷𝜷= 𝜷𝜷

= 𝜷𝜷

𝜸𝜸0𝜸𝜸1

𝜸𝜸4

⋮

= 𝜸𝜸= 𝜸𝜸

= 𝜸𝜸

1) Ψ𝜋𝜋,𝜎𝜎(𝑟𝑟) = 1, i.e., Ψ𝜋𝜋,𝜎𝜎 𝑟𝑟 = 𝑥𝑥 .2) For the unique 𝑥𝑥 ∈ Ψ𝜋𝜋,𝜎𝜎(𝑟𝑟), the pair of sequences 𝜷𝜷𝒙𝒙+𝒓𝒓 and 𝜸𝜸𝒙𝒙 is optimal.

Assume we have an optimal pair 𝜷𝜷,𝜸𝜸and a 𝟐𝟐 × 𝟏𝟏 circular Florentine array:

Then, any 𝑠𝑠 ∈ A 𝐵𝐵1,𝜋𝜋1and 𝑓𝑓 ∈ A 𝐵𝐵2,𝜋𝜋2 is an

optimal pair

Hong-Yeop Song

when 𝒎𝒎 > 𝟏𝟏

• Construct 𝑠𝑠 ∈ A 𝐵𝐵1,𝜋𝜋1 , 𝑓𝑓 ∈ A 𝐵𝐵2,𝜋𝜋2 with𝐵𝐵1 = 𝛽𝛽0,𝛽𝛽1, … ,𝛽𝛽𝑁𝑁−1 and 𝐵𝐵2 = 𝛾𝛾0, 𝛾𝛾1, … , 𝛾𝛾𝑁𝑁−1 , where

𝝅𝝅𝟏𝟏 0 1 2 3 4

𝝅𝝅𝟐𝟐 0 2 4 1 3

𝜷𝜷0𝜷𝜷1

𝜷𝜷4

⋮

= 𝜷𝜷= 𝜷𝜷

= 𝜷𝜷

𝜸𝜸0𝜸𝜸1

𝜸𝜸4

⋮

= 𝜸𝜸= 𝜸𝜸

= 𝜸𝜸

1) Ψ𝜋𝜋,𝜎𝜎(𝑟𝑟) = 1, i.e., Ψ𝜋𝜋,𝜎𝜎 𝑟𝑟 = 𝑥𝑥 .2) For the unique 𝑥𝑥 ∈ Ψ𝜋𝜋,𝜎𝜎(𝑟𝑟), the pair of sequences 𝜷𝜷𝒙𝒙+𝒓𝒓 and 𝜸𝜸𝒙𝒙 is optimal.

Assume we have an optimal pair 𝜷𝜷,𝜸𝜸and a 𝟐𝟐 × 𝟏𝟏 circular Florentine array:

𝜷𝜷0𝜷𝜷1

𝜷𝜷4

⋮

= 𝜸𝜸= 𝜷𝜷

= 𝜷𝜷

𝜸𝜸0𝜸𝜸1

𝜸𝜸4

⋮

= 𝜷𝜷= 𝜸𝜸

= 𝜸𝜸

Then, any 𝑠𝑠 ∈ A 𝐵𝐵1,𝜋𝜋1and 𝑓𝑓 ∈ A 𝐵𝐵2,𝜋𝜋2 is an

optimal pair

Hong-Yeop Song

𝜋𝜋1 0 1 2 3 4

𝜋𝜋2 0 2 4 1 3

Definition. Let 𝝅𝝅,𝝈𝝈 be two functions from ℤ𝑵𝑵 to ℤ𝒎𝒎𝑵𝑵. 𝜳𝜳𝝅𝝅,𝝈𝝈 𝒓𝒓 = 𝒙𝒙 ∈ ℤ𝑵𝑵 𝝅𝝅 𝒙𝒙 + 𝒓𝒓 ≡ 𝝈𝝈 𝒙𝒙 𝒎𝒎𝒎𝒎𝒎𝒎 𝑵𝑵 .

𝜳𝜳𝟏𝟏,𝟐𝟐 𝒓𝒓 = 𝒙𝒙 ∈ ℤ𝑵𝑵 𝝅𝝅𝟏𝟏 𝒙𝒙 + 𝒓𝒓 ≡ 𝝅𝝅𝟐𝟐 𝒙𝒙 𝒎𝒎𝒎𝒎𝒎𝒎 𝟏𝟏 ↔ 𝜷𝜷𝒙𝒙+𝒓𝒓 and 𝜸𝜸𝒙𝒙

Hong-Yeop Song

𝜋𝜋1 0 1 2 3 4

𝜋𝜋2 0 2 4 1 3

Definition. Let 𝝅𝝅,𝝈𝝈 be two functions from ℤ𝑵𝑵 to ℤ𝒎𝒎𝑵𝑵. 𝜳𝜳𝝅𝝅,𝝈𝝈 𝒓𝒓 = 𝒙𝒙 ∈ ℤ𝑵𝑵 𝝅𝝅 𝒙𝒙 + 𝒓𝒓 ≡ 𝝈𝝈 𝒙𝒙 𝒎𝒎𝒎𝒎𝒎𝒎 𝑵𝑵 .

𝛹𝛹1,2 𝟎𝟎 = 𝑥𝑥 ∈ ℤ𝑁𝑁 𝜋𝜋1 𝑥𝑥 + 𝟎𝟎 ≡ 𝜋𝜋2 𝑥𝑥 𝑚𝑚𝑚𝑚𝑑𝑑 5 = 0 ↔ 𝛽𝛽0+𝟎𝟎 = 𝛽𝛽𝟎𝟎 and 𝛾𝛾𝟎𝟎𝛹𝛹1,2 𝟏𝟏 = 𝑥𝑥 ∈ ℤ𝑁𝑁 𝜋𝜋1 𝑥𝑥 + 𝟏𝟏 ≡ 𝜋𝜋2 𝑥𝑥 𝑚𝑚𝑚𝑚𝑑𝑑 5 = 2 ↔ 𝛽𝛽2+𝟏𝟏 = 𝛽𝛽𝟑𝟑 and 𝛾𝛾𝟐𝟐𝛹𝛹1,2 𝟐𝟐 = 𝑥𝑥 ∈ ℤ𝑁𝑁 𝜋𝜋1 𝑥𝑥 + 𝟐𝟐 ≡ 𝜋𝜋2 𝑥𝑥 𝑚𝑚𝑚𝑚𝑑𝑑 5 = 4 ↔ 𝛽𝛽4+𝟐𝟐 = 𝛽𝛽𝟏𝟏 and 𝛾𝛾𝟒𝟒𝛹𝛹1,2 𝟑𝟑 = 𝑥𝑥 ∈ ℤ𝑁𝑁 𝜋𝜋1 𝑥𝑥 + 𝟑𝟑 ≡ 𝜋𝜋2 𝑥𝑥 𝑚𝑚𝑚𝑚𝑑𝑑 5 = 1 ↔ 𝛽𝛽1+𝟑𝟑 = 𝛽𝛽𝟒𝟒 and 𝛾𝛾𝟏𝟏𝛹𝛹1,2 𝟒𝟒 = 𝑥𝑥 ∈ ℤ𝑁𝑁 𝜋𝜋1 𝑥𝑥 + 𝟒𝟒 ≡ 𝜋𝜋2 𝑥𝑥 𝑚𝑚𝑚𝑚𝑑𝑑 5 = 3 ↔ 𝛽𝛽3+𝟒𝟒 = 𝛽𝛽𝟐𝟐 and 𝛾𝛾𝟑𝟑

𝜳𝜳𝟏𝟏,𝟐𝟐 𝒓𝒓 = 𝒙𝒙 ∈ ℤ𝑵𝑵 𝝅𝝅𝟏𝟏 𝒙𝒙 + 𝒓𝒓 ≡ 𝝅𝝅𝟐𝟐 𝒙𝒙 𝒎𝒎𝒎𝒎𝒎𝒎 𝟏𝟏 ↔ 𝜷𝜷𝒙𝒙+𝒓𝒓 and 𝜸𝜸𝒙𝒙

Hong-Yeop Song

𝜋𝜋1 0 1 2 3 4

𝜋𝜋2 0 2 4 1 3

Definition. Let 𝝅𝝅,𝝈𝝈 be two functions from ℤ𝑵𝑵 to ℤ𝒎𝒎𝑵𝑵. 𝜳𝜳𝝅𝝅,𝝈𝝈 𝒓𝒓 = 𝒙𝒙 ∈ ℤ𝑵𝑵 𝝅𝝅 𝒙𝒙 + 𝒓𝒓 ≡ 𝝈𝝈 𝒙𝒙 𝒎𝒎𝒎𝒎𝒎𝒎 𝑵𝑵 .

𝛹𝛹1,2 𝟎𝟎 = 𝑥𝑥 ∈ ℤ𝑁𝑁 𝜋𝜋1 𝑥𝑥 + 𝟎𝟎 ≡ 𝜋𝜋2 𝑥𝑥 𝑚𝑚𝑚𝑚𝑑𝑑 5 = 0 ↔ 𝛽𝛽0+𝟎𝟎 = 𝛽𝛽𝟎𝟎 and 𝛾𝛾𝟎𝟎𝛹𝛹1,2 𝟏𝟏 = 𝑥𝑥 ∈ ℤ𝑁𝑁 𝜋𝜋1 𝑥𝑥 + 𝟏𝟏 ≡ 𝜋𝜋2 𝑥𝑥 𝑚𝑚𝑚𝑚𝑑𝑑 5 = 2 ↔ 𝛽𝛽2+𝟏𝟏 = 𝛽𝛽𝟑𝟑 and 𝛾𝛾𝟐𝟐𝛹𝛹1,2 𝟐𝟐 = 𝑥𝑥 ∈ ℤ𝑁𝑁 𝜋𝜋1 𝑥𝑥 + 𝟐𝟐 ≡ 𝜋𝜋2 𝑥𝑥 𝑚𝑚𝑚𝑚𝑑𝑑 5 = 4 ↔ 𝛽𝛽4+𝟐𝟐 = 𝛽𝛽𝟏𝟏 and 𝛾𝛾𝟒𝟒𝛹𝛹1,2 𝟑𝟑 = 𝑥𝑥 ∈ ℤ𝑁𝑁 𝜋𝜋1 𝑥𝑥 + 𝟑𝟑 ≡ 𝜋𝜋2 𝑥𝑥 𝑚𝑚𝑚𝑚𝑑𝑑 5 = 1 ↔ 𝛽𝛽1+𝟑𝟑 = 𝛽𝛽𝟒𝟒 and 𝛾𝛾𝟏𝟏𝛹𝛹1,2 𝟒𝟒 = 𝑥𝑥 ∈ ℤ𝑁𝑁 𝜋𝜋1 𝑥𝑥 + 𝟒𝟒 ≡ 𝜋𝜋2 𝑥𝑥 𝑚𝑚𝑚𝑚𝑑𝑑 5 = 3 ↔ 𝛽𝛽3+𝟒𝟒 = 𝛽𝛽𝟐𝟐 and 𝛾𝛾𝟑𝟑

𝜳𝜳𝟏𝟏,𝟐𝟐 𝒓𝒓 = 𝒙𝒙 ∈ ℤ𝑵𝑵 𝝅𝝅𝟏𝟏 𝒙𝒙 + 𝒓𝒓 ≡ 𝝅𝝅𝟐𝟐 𝒙𝒙 𝒎𝒎𝒎𝒎𝒎𝒎 𝟏𝟏 ↔ 𝜷𝜷𝒙𝒙+𝒓𝒓 and 𝜸𝜸𝒙𝒙

(𝜷𝜷0,

(𝜷𝜷4,

𝜸𝜸0) =

𝜸𝜸1) =

(𝜷𝜷,𝜸𝜸) or (𝜸𝜸,𝜷𝜷)

(𝜷𝜷3, 𝜸𝜸2) =

(𝜷𝜷,𝜸𝜸) or (𝜸𝜸,𝜷𝜷)(𝜷𝜷,𝜸𝜸) or (𝜸𝜸,𝜷𝜷)(𝜷𝜷,𝜸𝜸) or (𝜸𝜸,𝜷𝜷)(𝜷𝜷,𝜸𝜸) or (𝜸𝜸,𝜷𝜷)

(𝜷𝜷2, 𝜸𝜸3) =(𝜷𝜷1, 𝜸𝜸4) =

Hong-Yeop Song

Maximum set size

• Let 𝑭𝑭𝒄𝒄 𝑵𝑵 be the maximum such that an 𝑭𝑭𝒄𝒄 𝑵𝑵 × 𝑵𝑵circular Florentine array exists.

• Let 𝑶𝑶𝑮𝑮(𝒎𝒎𝑵𝑵𝟐𝟐) be the maximum such that an optimal 𝑶𝑶𝑮𝑮(𝒎𝒎𝑵𝑵𝟐𝟐)-set of generalized Milewski sequences of length 𝒎𝒎𝑵𝑵𝟐𝟐 from perfect sequences of length 𝒎𝒎.

Hong-Yeop Song

Maximum set size

• Let 𝑭𝑭𝒄𝒄 𝑵𝑵 be the maximum such that an 𝑭𝑭𝒄𝒄 𝑵𝑵 × 𝑵𝑵circular Florentine array exists.

• Let 𝑶𝑶𝑮𝑮(𝒎𝒎𝑵𝑵𝟐𝟐) be the maximum such that an optimal 𝑶𝑶𝑮𝑮(𝒎𝒎𝑵𝑵𝟐𝟐)-set of generalized Milewski sequences of length 𝒎𝒎𝑵𝑵𝟐𝟐 from perfect sequences of length 𝒎𝒎.

1) When 𝑚𝑚 = 1,𝑂𝑂𝐺𝐺 mN2 = 𝑂𝑂𝐺𝐺 N2 = 𝐹𝐹𝑐𝑐 𝑁𝑁 .

Hong-Yeop Song

Maximum set size

• Let 𝑭𝑭𝒄𝒄 𝑵𝑵 be the maximum such that an 𝑭𝑭𝒄𝒄 𝑵𝑵 × 𝑵𝑵circular Florentine array exists.

• Let 𝑶𝑶𝑮𝑮(𝒎𝒎𝑵𝑵𝟐𝟐) be the maximum such that an optimal 𝑶𝑶𝑮𝑮(𝒎𝒎𝑵𝑵𝟐𝟐)-set of generalized Milewski sequences of length 𝒎𝒎𝑵𝑵𝟐𝟐 from perfect sequences of length 𝒎𝒎.

1) When 𝑚𝑚 = 1,𝑂𝑂𝐺𝐺 mN2 = 𝑂𝑂𝐺𝐺 N2 = 𝐹𝐹𝑐𝑐 𝑁𝑁 .

2) When 𝑚𝑚 ≥ 2, 𝑂𝑂𝐺𝐺 mN2 = min 𝑂𝑂𝑃𝑃 𝑚𝑚 ,𝐹𝐹𝑐𝑐 𝑁𝑁

where 𝑂𝑂𝑃𝑃 𝑚𝑚 is the maximum such that an optimal 𝑂𝑂𝑃𝑃 𝑚𝑚 -set of perfect sequences of period 𝑚𝑚.

Hong-Yeop Song

Concluding remarks

• To obtain an optimal 𝒌𝒌-set of generalized Milewski sequences of length 𝒎𝒎𝑵𝑵𝟐𝟐, we need both:▫ A 𝒌𝒌 × 𝑵𝑵 circular Florentine array, and▫ An optimal 𝒌𝒌-set of perfect sequences of length 𝒎𝒎.

(When 𝑚𝑚 = 1, a trivial example will work always)

Hong-Yeop Song

Concluding remarks

• To obtain an optimal 𝒌𝒌-set of generalized Milewski sequences of length 𝒎𝒎𝑵𝑵𝟐𝟐, we need both:▫ A 𝒌𝒌 × 𝑵𝑵 circular Florentine array, and▫ An optimal 𝒌𝒌-set of perfect sequences of length 𝒎𝒎.

(When 𝑚𝑚 = 1, a trivial example will work always)

Some open problems:• Find any other positive odd integer 𝑵𝑵 such that 𝑭𝑭𝒄𝒄(𝑵𝑵)

is greater than 𝒑𝒑𝒎𝒎𝒎𝒎𝒏𝒏 − 𝟏𝟏.• Determine the exact value of 𝑭𝑭𝒄𝒄(𝑵𝑵) for every odd 𝑵𝑵.

Hong-Yeop Song

Thanks !