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Quaternary Golay Sequence Pairs Richard Gibson Department of Mathematics Simon Fraser University Masters Thesis Defence November 6, 2008 Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 1 / 33 Outline 1 Background Quaternary Golay

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SLIDE 1

Quaternary Golay Sequence Pairs

Richard Gibson

Department of Mathematics Simon Fraser University

Masters Thesis Defence November 6, 2008

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 1 / 33

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SLIDE 2

Outline

1 Background

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 2 / 33

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SLIDE 3

Outline

1 Background 2 Classifying Quaternary Golay Sequence Pairs

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 2 / 33

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SLIDE 4

Outline

1 Background 2 Classifying Quaternary Golay Sequence Pairs 3 Constructing a Binary Barker Sequence from a Quaternary Golay

Sequence

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 2 / 33

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SLIDE 5

Outline

1 Background 2 Classifying Quaternary Golay Sequence Pairs 3 Constructing a Binary Barker Sequence from a Quaternary Golay

Sequence

4 Summary and Open Problems

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 2 / 33

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SLIDE 6

Background

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 3 / 33

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SLIDE 7

Sequences

Example

1 2 3 quaternary sequence

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 4 / 33

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SLIDE 8

Sequences

Example

1 2 3 quaternary sequence A = (a0, ..., an−1) is a quaternary sequence if aj ∈ Z4 for all 0 ≤ j < n.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 4 / 33

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SLIDE 9

Sequences

Example

1 2 3 quaternary sequence A = (a0, ..., an−1) is a quaternary sequence if aj ∈ Z4 for all 0 ≤ j < n.

Example

2 2 2 2 binary sequence

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 4 / 33

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SLIDE 10

Sequences

Example

1 2 3 quaternary sequence A = (a0, ..., an−1) is a quaternary sequence if aj ∈ Z4 for all 0 ≤ j < n.

Example

2 2 2 2 binary sequence A = (a0, ..., an−1) is a binary sequence if aj ∈ {0, 2} for all 0 ≤ j < n.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 4 / 33

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SLIDE 11

Aperiodic autocorrelation function

Define CA(u), the aperiodic autocorrelation function of the quaternary sequence A at shift u, by: CA(u) =

n−u−1

  • j=0

iaj−aj+u

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 5 / 33

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SLIDE 12

Aperiodic autocorrelation function

Define CA(u), the aperiodic autocorrelation function of the quaternary sequence A at shift u, by: A = 1 2 3 CA(u) =

n−u−1

  • j=0

iaj−aj+u

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 5 / 33

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SLIDE 13

Aperiodic autocorrelation function

Define CA(u), the aperiodic autocorrelation function of the quaternary sequence A at shift u, by: A = 1 2 3 A = 1 2 3 CA(u) =

n−u−1

  • j=0

iaj−aj+u

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 5 / 33

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SLIDE 14

Aperiodic autocorrelation function

Define CA(u), the aperiodic autocorrelation function of the quaternary sequence A at shift u, by: A = 1 2 3

  • A =

1 2 3 CA(1) = i0−1 + i1−2 + i2−0 + i0−3 = −1 − i (where i = √ −1) CA(u) =

n−u−1

  • j=0

iaj−aj+u

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 5 / 33

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SLIDE 15

Aperiodic autocorrelation function

Define CA(u), the aperiodic autocorrelation function of the quaternary sequence A at shift u, by: A = 1 2 3 A = 1 2 3

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 6 / 33

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SLIDE 16

Aperiodic autocorrelation function

Define CA(u), the aperiodic autocorrelation function of the quaternary sequence A at shift u, by: A = 1 2 3

  • A =

1 2 3 CA(2) = i0−2 + i1−0 + i2−3 = −1

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 6 / 33

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SLIDE 17

Aperiodic autocorrelation function

Define CA(u), the aperiodic autocorrelation function of the quaternary sequence A at shift u, by: A = 1 2 3

  • A =

1 2 3 CA(2) = i0−2 + i1−0 + i2−3 = −1 CA(u) :=

n−u−1

  • j=0

iaj−aj+u for all 0 ≤ u < n

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 6 / 33

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SLIDE 18

What is a Barker sequence?

A quaternary (binary) sequence A of length n is a quaternary (binary) Barker sequence if |CA(u)| ∈ {0, 1} for all 1 ≤ u < n.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 7 / 33

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SLIDE 19

What is a Barker sequence?

A quaternary (binary) sequence A of length n is a quaternary (binary) Barker sequence if |CA(u)| ∈ {0, 1} for all 1 ≤ u < n.

Example

A = 2 2 2 2 u 1 2 3 4 5 6 7 8 9 10 11 12 |CA(u)| 13 1 1 1 1 1 1

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 7 / 33

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SLIDE 20

What is a Barker sequence?

A quaternary (binary) sequence A of length n is a quaternary (binary) Barker sequence if |CA(u)| ∈ {0, 1} for all 1 ≤ u < n.

Example

A = 2 2 2 2 u 1 2 3 4 5 6 7 8 9 10 11 12 |CA(u)| 13 1 1 1 1 1 1 Applications include:

  • radar
  • pulse compression

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 7 / 33

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SLIDE 21

History of Barker sequences

  • Binary Barker sequences exist for lengths 2, 3, 4, 5, 7, 11, and 13.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 8 / 33

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SLIDE 22

History of Barker sequences

  • Binary Barker sequences exist for lengths 2, 3, 4, 5, 7, 11, and 13.
  • There are no binary Barker sequences of odd length > 13 (Turyn and

Storer, 1961).

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 8 / 33

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SLIDE 23

History of Barker sequences

  • Binary Barker sequences exist for lengths 2, 3, 4, 5, 7, 11, and 13.
  • There are no binary Barker sequences of odd length > 13 (Turyn and

Storer, 1961).

  • Barker Sequence Conjecture: There are no binary Barker sequences of

length > 13.

  • smallest open case is for length > 1022 (Leung and Schmidt, 2005).

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 8 / 33

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SLIDE 24

What is a Golay sequence?

Let A and B be quaternary (binary) sequences of length n.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 9 / 33

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SLIDE 25

What is a Golay sequence?

Let A and B be quaternary (binary) sequences of length n. A sequence pair (A, B) is a quaternary (binary) Golay sequence pair if CA(u) + CB(u) = 0 for all 1 ≤ u < n (we say A and B are quaternary (binary) Golay sequences).

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 9 / 33

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SLIDE 26

What is a Golay sequence?

Let A and B be quaternary (binary) sequences of length n. A sequence pair (A, B) is a quaternary (binary) Golay sequence pair if CA(u) + CB(u) = 0 for all 1 ≤ u < n (we say A and B are quaternary (binary) Golay sequences).

Example

A = 1 2 3 B = 3 1 u 1 2 3 4 CA(u) 5 −1 − i −1 i CB(u) 5 1 + i 1 −i CA(u) + CB(u) 10

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 9 / 33

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SLIDE 27

What is a Golay sequence?

Let A and B be quaternary (binary) sequences of length n. A sequence pair (A, B) is a quaternary (binary) Golay sequence pair if CA(u) + CB(u) = 0 for all 1 ≤ u < n (we say A and B are quaternary (binary) Golay sequences).

Example

A = 1 2 3 B = 3 1 u 1 2 3 4 CA(u) 5 −1 − i −1 i CB(u) 5 1 + i 1 −i CA(u) + CB(u) 10 Applications include:

  • medical ultrasound, etc.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 9 / 33

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SLIDE 28

History of Golay sequences

  • There exist binary Golay pairs of length 2a10b26c for all integers

a, b, c ≥ 0 (Turyn, 1974).

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 10 / 33

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SLIDE 29

History of Golay sequences

  • There exist binary Golay pairs of length 2a10b26c for all integers

a, b, c ≥ 0 (Turyn, 1974).

  • All binary pairs of length less than 100 have been classified (Borwein

and Ferguson, 2003).

  • i.e. each pair either derivable from some general construction method
  • r identified as one of five “seed” pair

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 10 / 33

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SLIDE 30

History of Golay sequences

  • There exist binary Golay pairs of length 2a10b26c for all integers

a, b, c ≥ 0 (Turyn, 1974).

  • All binary pairs of length less than 100 have been classified (Borwein

and Ferguson, 2003).

  • i.e. each pair either derivable from some general construction method
  • r identified as one of five “seed” pair
  • Multi-dimensional construction process: Can obtain Golay pairs from

a series of shorter Golay pairs (Fiedler, Jedwab and Parker, 2008).

  • Explains all known binary and quaternary Golay sequences of length 2m.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 10 / 33

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SLIDE 31

History of Golay sequences

  • There exist binary Golay pairs of length 2a10b26c for all integers

a, b, c ≥ 0 (Turyn, 1974).

  • All binary pairs of length less than 100 have been classified (Borwein

and Ferguson, 2003).

  • i.e. each pair either derivable from some general construction method
  • r identified as one of five “seed” pair
  • Multi-dimensional construction process: Can obtain Golay pairs from

a series of shorter Golay pairs (Fiedler, Jedwab and Parker, 2008).

  • Explains all known binary and quaternary Golay sequences of length 2m.

There are only two nonexistence results for Golay sequences:

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 10 / 33

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SLIDE 32

History of Golay sequences

  • There exist binary Golay pairs of length 2a10b26c for all integers

a, b, c ≥ 0 (Turyn, 1974).

  • All binary pairs of length less than 100 have been classified (Borwein

and Ferguson, 2003).

  • i.e. each pair either derivable from some general construction method
  • r identified as one of five “seed” pair
  • Multi-dimensional construction process: Can obtain Golay pairs from

a series of shorter Golay pairs (Fiedler, Jedwab and Parker, 2008).

  • Explains all known binary and quaternary Golay sequences of length 2m.

There are only two nonexistence results for Golay sequences:

  • Binary Golay sequences must have even length (Marcel Golay, 1961).

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 10 / 33

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SLIDE 33

History of Golay sequences

  • There exist binary Golay pairs of length 2a10b26c for all integers

a, b, c ≥ 0 (Turyn, 1974).

  • All binary pairs of length less than 100 have been classified (Borwein

and Ferguson, 2003).

  • i.e. each pair either derivable from some general construction method
  • r identified as one of five “seed” pair
  • Multi-dimensional construction process: Can obtain Golay pairs from

a series of shorter Golay pairs (Fiedler, Jedwab and Parker, 2008).

  • Explains all known binary and quaternary Golay sequences of length 2m.

There are only two nonexistence results for Golay sequences:

  • Binary Golay sequences must have even length (Marcel Golay, 1961).
  • The length of a binary Golay pair has no prime factor congruent to 3

modulo 4 (Eliahou, Kervaire and Saffari, 1991).

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 10 / 33

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SLIDE 34

History of Golay sequences

  • There exist binary Golay pairs of length 2a10b26c for all integers

a, b, c ≥ 0 (Turyn, 1974).

  • All binary pairs of length less than 100 have been classified (Borwein

and Ferguson, 2003).

  • i.e. each pair either derivable from some general construction method
  • r identified as one of five “seed” pair
  • Multi-dimensional construction process: Can obtain Golay pairs from

a series of shorter Golay pairs (Fiedler, Jedwab and Parker, 2008).

  • Explains all known binary and quaternary Golay sequences of length 2m.

There are only two nonexistence results for Golay sequences:

  • Binary Golay sequences must have even length (Marcel Golay, 1961).
  • The length of a binary Golay pair has no prime factor congruent to 3

modulo 4 (Eliahou, Kervaire and Saffari, 1991). No nonexistence results for quaternary Golay sequences.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 10 / 33

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SLIDE 35

Classifying Quaternary Golay Sequence Pairs

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 11 / 33

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SLIDE 36

Ordered quaternary Golay sequence pair counts

In 2002, Craigen, Holzmann, and Kharaghani exhaustively found all

  • rdered quaternary Golay sequence pairs of small length n:

n

# pairs

n

# pairs

n

# pairs

1 16 8 6656 15 2 64 9 16 106496 3 128 10 12288 17 4 512 11 512 18 24576 5 512 12 36864 19 6 2048 13 512 20 215040∗ 7 14 21

∗ Frank Fiedler, personal communication.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 12 / 33

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SLIDE 37

Ordered quaternary Golay sequence pair counts

In 2002, Craigen, Holzmann, and Kharaghani exhaustively found all

  • rdered quaternary Golay sequence pairs of small length n:

n

# pairs

n

# pairs

n

# pairs

1 16 8 6656 15 2 64 9 16 106496 3 128 10 12288 17 4 512 11 512 18 24576 5 512 12 36864 19 6 2048 13 512 20 215040∗ 7 14 21

  • Using small “seed” pairs, how many of the pairs in the table above

can be explained using the multi-dimensional construction process and other constructions?

∗ Frank Fiedler, personal communication.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 12 / 33

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SLIDE 38

Ordered quaternary Golay sequence pair counts

In 2002, Craigen, Holzmann, and Kharaghani exhaustively found all

  • rdered quaternary Golay sequence pairs of small length n:

n

# pairs

n

# pairs

n

# pairs

1 16 8 6656 15 2 64 9 16 106496 3 128 10 12288 17 4 512 11 512 18 24576 5 512 12 36864 19 6 2048 13 512 20 215040∗ 7 14 21

  • Using small “seed” pairs, how many of the pairs in the table above

can be explained using the multi-dimensional construction process and other constructions?

  • How can we explain the existence of the seed pairs?

∗ Frank Fiedler, personal communication.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 12 / 33

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SLIDE 39

Multi-dimensional construction process

How do we use the multi-dimensional construction process?

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 13 / 33

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SLIDE 40

Multi-dimensional construction process

How do we use the multi-dimensional construction process?

  • Input: m + 1 Golay sequence pairs of length n0, n1, ...nm, where

m ≥ 1, to create a multi-dimensional object.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 13 / 33

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SLIDE 41

Multi-dimensional construction process

How do we use the multi-dimensional construction process?

  • Input: m + 1 Golay sequence pairs of length n0, n1, ...nm, where

m ≥ 1, to create a multi-dimensional object.

  • Process multi-dimensional object...

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 13 / 33

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SLIDE 42

Multi-dimensional construction process

How do we use the multi-dimensional construction process?

  • Input: m + 1 Golay sequence pairs of length n0, n1, ...nm, where

m ≥ 1, to create a multi-dimensional object.

  • Process multi-dimensional object...
  • Output: A collection of Golay sequence pairs, all of length

n0 · n1 · ... · nm · 2m.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 13 / 33

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SLIDE 43

Let’s use the multi-dimensional construction process

We can construct ordered quaternary Golay sequence pairs using “seed” Golay pairs of length 1, 3 and 5 as follows:

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 14 / 33

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SLIDE 44

Let’s use the multi-dimensional construction process

We can construct ordered quaternary Golay sequence pairs using “seed” Golay pairs of length 1, 3 and 5 as follows:

  • ,
  • →

64 ordered pairs of length 1 · 1 · 2 = 2

  • ,
  • Quaternary Golay Sequence Pairs : Richard Gibson (SFU)

14 / 33

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SLIDE 45

Ordered quaternary Golay sequence pair counts

In 2002, Craigen, Holzmann, and Kharaghani exhaustively found all

  • rdered quaternary Golay sequence pairs of small length n:

n

# pairs

n

# pairs

n

# pairs

1 16 8 6656 15 2 64 9 16 106496 3 128 10 12288 17 4 512 11 512 18 24576 5 512 12 36864 19 6 2048 13 512 20 215040∗ 7 14 21

∗ Frank Fiedler, personal communication.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 15 / 33

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SLIDE 46

Let’s use the multi-dimensional construction process

We can construct ordered quaternary Golay sequence pairs using “seed” Golay pairs of length 1, 3 and 5 as follows:

  • ,
  • →

64 ordered pairs of length 1 · 1 · 2 = 2

  • ,
  • Quaternary Golay Sequence Pairs : Richard Gibson (SFU)

16 / 33

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SLIDE 47

Let’s use the multi-dimensional construction process

We can construct ordered quaternary Golay sequence pairs using “seed” Golay pairs of length 1, 3 and 5 as follows:

  • ,
  • →

64 ordered pairs of length 1 · 1 · 2 = 2

  • ,
  • ,
  • →

512 ordered pairs of length 1 · 1 · 1 · 22 = 4

  • ,
  • ,
  • Quaternary Golay Sequence Pairs : Richard Gibson (SFU)

16 / 33

slide-48
SLIDE 48

Ordered quaternary Golay sequence pair counts

In 2002, Craigen, Holzmann, and Kharaghani exhaustively found all

  • rdered quaternary Golay sequence pairs of small length n:

n

# pairs

n

# pairs

n

# pairs

1 16 8 6656 15 2 64 9 16 106496 3 128 10 12288 17 4 512 11 512 18 24576 5 512 12 36864 19 6 2048 13 512 20 215040∗ 7 14 21

∗ Frank Fiedler, personal communication.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 17 / 33

slide-49
SLIDE 49

Let’s use the multi-dimensional construction process

We can construct ordered quaternary Golay sequence pairs using “seed” Golay pairs of length 1, 3 and 5 as follows:

  • ,
  • →

64 ordered pairs of length 1 · 1 · 2 = 2

  • ,
  • ,
  • →

512 ordered pairs of length 1 · 1 · 1 · 22 = 4

  • ,
  • ,
  • Quaternary Golay Sequence Pairs : Richard Gibson (SFU)

18 / 33

slide-50
SLIDE 50

Let’s use the multi-dimensional construction process

We can construct ordered quaternary Golay sequence pairs using “seed” Golay pairs of length 1, 3 and 5 as follows:

  • ,
  • →

64 ordered pairs of length 1 · 1 · 2 = 2

  • ,
  • ,
  • →

512 ordered pairs of length 1 · 1 · 1 · 22 = 4

  • ,
  • ,
  • 2

, 1

  • →

2048 ordered pairs of length 3 · 1 · 2 = 6

  • ,
  • Quaternary Golay Sequence Pairs : Richard Gibson (SFU)

18 / 33

slide-51
SLIDE 51

Let’s use the multi-dimensional construction process

We can construct ordered quaternary Golay sequence pairs using “seed” Golay pairs of length 1, 3 and 5 as follows:

  • ,
  • →

64 ordered pairs of length 1 · 1 · 2 = 2

  • ,
  • ,
  • →

512 ordered pairs of length 1 · 1 · 1 · 22 = 4

  • ,
  • ,
  • 2

, 1

  • →

2048 ordered pairs of length 3 · 1 · 2 = 6

  • ,
  • ,
  • →

6144 ordered pairs of length 1 · 1 · 1 · 1 · 23 = 8

  • ,
  • ,
  • ,
  • Quaternary Golay Sequence Pairs : Richard Gibson (SFU)

18 / 33

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SLIDE 52

Let’s use the multi-dimensional construction process

Continue as follows:

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 19 / 33

slide-53
SLIDE 53

Let’s use the multi-dimensional construction process

Continue as follows:

  • 1

2 3 , 3 1

  • →

8192 pairs of length 10

  • ,
  • 2

, 1

  • →

36864 pairs of length 12

  • ,
  • ,
  • ,
  • →

98304 pairs of length 16

  • ,
  • ,
  • ,
  • ,
  • Quaternary Golay Sequence Pairs : Richard Gibson (SFU)

19 / 33

slide-54
SLIDE 54

Let’s use the multi-dimensional construction process

Coninue as follows:

  • 2

, 1

  • →

24576 pairs of length 18

  • 2

, 1

  • 1

2 3 , 3 1

  • →

147456 pairs of length 20

  • ,
  • ,
  • Quaternary Golay Sequence Pairs : Richard Gibson (SFU)

20 / 33

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SLIDE 55

Ordered quaternary Golay pairs left to explain

n

# pairs # pairs

n

# pairs # pairs

n

# pairs # pairs

left to left to left to explain explain explain 1 16 8 6656 512 15 2 64 9 16 106496 8192 3 128 128 10 12288 4096 17 4 512 11 512 512 18 24576 5 512 512 12 36864 19 6 2048 13 512 512 20 215040 67584 7 14 21

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 21 / 33

slide-56
SLIDE 56

Ordered quaternary Golay pairs left to explain

n

# pairs # pairs

n

# pairs # pairs

n

# pairs # pairs

left to left to left to explain explain explain 1 16 8 6656 512 15 2 64 9 16 106496 8192 3 128 128 10 12288 4096 17 4 512 11 512 512 18 24576 5 512 512 12 36864 19 6 2048 13 512 512 20 215040 67584 7 14 21

  • “Shared autocorrelation property”

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 21 / 33

slide-57
SLIDE 57

Ordered quaternary Golay pairs left to explain

n

# pairs # pairs

n

# pairs # pairs

n

# pairs # pairs

left to left to left to explain explain explain 1 16 8 6656 512 15 2 64 9 16 106496 8192 3 128 128 10 12288 4096 17 4 512 11 512 512 18 24576 5 512 512 12 36864 19 6 2048 13 512 512 20 215040 67584 7 14 21

  • “Shared autocorrelation property”
  • Multi-dimensional construction process with special length 8 pairs and a

trivial length 1 pair

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 21 / 33

slide-58
SLIDE 58

Ordered quaternary Golay pairs left to explain

n

# pairs # pairs

n

# pairs # pairs

n

# pairs # pairs

left to left to left to explain explain explain 1 16 8 6656 15 2 64 9 16 106496 3 128 128 10 12288 4096 17 4 512 11 512 512 18 24576 5 512 512 12 36864 19 6 2048 13 512 512 20 215040 67584 7 14 21

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 21 / 33

slide-59
SLIDE 59

Ordered quaternary Golay pairs left to explain

n

# pairs # pairs

n

# pairs # pairs

n

# pairs # pairs

left to left to left to explain explain explain 1 16 8 6656 15 2 64 9 16 106496 3 128 128 10 12288 4096 17 4 512 11 512 512 18 24576 5 512 512 12 36864 19 6 2048 13 512 512 20 215040 67584 7 14 21

  • Symmetry Lemma: (A, B) are a Golay pair ⇔ (A + B, A − B) are a Golay

pair (where A and B are in “multiplicative” notation).

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 21 / 33

slide-60
SLIDE 60

Ordered quaternary Golay pairs left to explain

n

# pairs # pairs

n

# pairs # pairs

n

# pairs # pairs

left to left to left to explain explain explain 1 16 8 6656 15 2 64 9 16 106496 3 128 128 10 12288 4096 17 4 512 11 512 512 18 24576 5 512 512 12 36864 19 6 2048 13 512 512 20 215040 67584 7 14 21

  • Symmetry Lemma: (A, B) are a Golay pair ⇔ (A + B, A − B) are a Golay

pair (where A and B are in “multiplicative” notation).

  • Explains all remaining length 10 pairs and 2048 of the remaining length 20

pairs from binary Golay seed pairs.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 21 / 33

slide-61
SLIDE 61

Ordered quaternary Golay pairs left to explain

n

# pairs # pairs

n

# pairs # pairs

n

# pairs # pairs

left to left to left to explain explain explain 1 16 8 6656 15 2 64 9 16 106496 3 128 128 10 12288 17 4 512 11 512 512 18 24576 5 512 512 12 36864 19 6 2048 13 512 512 20 215040 65536 7 14 21

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 21 / 33

slide-62
SLIDE 62

Ordered quaternary Golay pairs left to explain

n

# pairs # pairs

n

# pairs # pairs

n

# pairs # pairs

left to left to left to explain explain explain 1 16 8 6656 15 2 64 9 16 106496 3 128 128 10 12288 17 4 512 11 512 512 18 24576 5 512 512 12 36864 19 6 2048 13 512 512 20 215040 65536 7 14 21

  • Multi-dimensional construction process with special length 10 pairs and a

trivial length 1 pair

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 21 / 33

slide-63
SLIDE 63

Ordered quaternary Golay pairs left to explain

n

# pairs # pairs

n

# pairs # pairs

n

# pairs # pairs

left to left to left to explain explain explain 1 16 8 6656 15 2 64 9 16 106496 3 128 128 10 12288 17 4 512 11 512 512 18 24576 5 512 512 12 36864 19 6 2048 13 512 512 20 215040 7 14 21

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 21 / 33

slide-64
SLIDE 64

Ordered quaternary Golay pairs left to explain

n

# pairs # pairs

n

# pairs # pairs

n

# pairs # pairs

left to left to left to explain explain explain 1 16 8 6656 15 2 64 9 16 106496 3 128 128 10 12288 17 4 512 11 512 512 18 24576 5 512 512 12 36864 19 6 2048 13 512 512 20 215040 7 14 21 Let’s look at lengths 5 and 13...

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 21 / 33

slide-65
SLIDE 65

Lengths 5 and 13

G5,1 = 3 1 G5,2 = 1 2 3 quaternary Golay pair of length 5 G13,1 = 1 2 3 2 3 1 G13,2 = 1 2 2 2 1 2 3 2 3 quaternary Golay pair of length 13

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 22 / 33

slide-66
SLIDE 66

Lengths 5 and 13

G5,1 = 3 1 G5,2 = 1 2 3 quaternary Golay pair of length 5 G13,1 = 1 2 3 2 3 1 G13,2 = 1 2 2 2 1 2 3 2 3 quaternary Golay pair of length 13 B5 = 2 binary Barker sequence of length 5 B13 = 2 2 2 2 binary Barker sequence of length 13

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 22 / 33

slide-67
SLIDE 67

Lengths 5 and 13 (cont.)

B5 + G5,1 = 1 1 = int(W0, X0) B5 + G5,2 = 1 2 2 3 = int(Y0, Z0)

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 23 / 33

slide-68
SLIDE 68

Lengths 5 and 13 (cont.)

B5 + G5,1 = 1 1 = int(W0, X0) B5 + G5,2 = 1 2 2 3 = int(Y0, Z0) B13 + G13,1 = 1 2 2 2 3 1 1 = int(W1, X1) B13 + G13,2 = 1 2 2 2 3 1 2 2 3 = int(Y1, Z1)

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 23 / 33

slide-69
SLIDE 69

Lengths 5 and 13 (cont.)

B5 + G5,1 = 1 1 = int(W0, X0) B5 + G5,2 = 1 2 2 3 = int(Y0, Z0) B13 + G13,1 = 1 2 2 2 3 1 1 = int(W1, X1) B13 + G13,2 = 1 2 2 2 3 1 2 2 3 = int(Y1, Z1)

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 23 / 33

slide-70
SLIDE 70

Lengths 5 and 13 (cont.)

B5 + G5,1 = 1 1 = int(W0, X0) B5 + G5,2 = 1 2 2 3 = int(Y0, Z0) B13 + G13,1 = 1 2 2 2 3 1 1 = int(W1, X1) B13 + G13,2 = 1 2 2 2 3 1 2 2 3 = int(Y1, Z1)

Xm := ((0 1 2 3)m 0 1) Zm := ((1 2 3 0)m 1 2)

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 23 / 33

slide-71
SLIDE 71

Lengths 5 and 13 (cont.)

B5 + G5,1 = 1 1 = int(W0, X0) B5 + G5,2 = 1 2 2 3 = int(Y0, Z0) B13 + G13,1 = 1 2 2 2 3 1 1 = int(W1, X1) B13 + G13,2 = 1 2 2 2 3 1 2 2 3 = int(Y1, Z1)

Xm := ((0 1 2 3)m 0 1) Zm := ((1 2 3 0)m 1 2) Wm := ((0 0 2 2)m 0 0 1) Ym := ((0 2 2 0)m 0 2 3)

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 23 / 33

slide-72
SLIDE 72

Lengths 5 and 13 (cont.)

B5 + G5,1 = 1 1 = int(W0, X0) B5 + G5,2 = 1 2 2 3 = int(Y0, Z0) B13 + G13,1 = 1 2 2 2 3 1 1 = int(W1, X1) B13 + G13,2 = 1 2 2 2 3 1 2 2 3 = int(Y1, Z1)

Xm := ((0 1 2 3)m 0 1) Zm := ((1 2 3 0)m 1 2) Wm := ((0 0 2 2)m 0 0 1) Ym := ((0 2 2 0)m 0 2 3)

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 23 / 33

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SLIDE 73

Binary Barker to quaternary Golay

Theorem

Let m ∈ N. Suppose int(A, B) is a binary Barker sequence of length 8m + 5 where A = ((0 0 0 2)m 0 0 0). Then the sequences E := int(A + Wm, B + Xm), F := int(A + Ym, B + Zm) form a quaternary Golay pair of length 8m + 5.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 24 / 33

slide-74
SLIDE 74

Binary Barker to quaternary Golay

Theorem

Let m ∈ N. Suppose int(A, B) is a binary Barker sequence of length 8m + 5 where A = ((0 0 0 2)m 0 0 0). Then the sequences E := int(A + Wm, B + Xm), F := int(A + Ym, B + Zm) form a quaternary Golay pair of length 8m + 5.

  • Explains all length 5 and 13 quaternary Golay pairs!

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 24 / 33

slide-75
SLIDE 75

Binary Barker to quaternary Golay

Theorem

Let m ∈ N. Suppose int(A, B) is a binary Barker sequence of length 8m + 5 where A = ((0 0 0 2)m 0 0 0). Then the sequences E := int(A + Wm, B + Xm), F := int(A + Ym, B + Zm) form a quaternary Golay pair of length 8m + 5.

  • Explains all length 5 and 13 quaternary Golay pairs!
  • Unfortunately, this result does not give rise to any new quaternary

Golay pairs.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 24 / 33

slide-76
SLIDE 76

Ordered quaternary Golay pairs left to explain

n

# pairs # pairs

n

# pairs # pairs

n

# pairs # pairs

left to left to left to explain explain explain 1 16 8 6656 15 2 64 9 16 106496 3 128 128 10 12288 17 4 512 11 512 512 18 24576 5 512 512 12 36864 19 6 2048 13 512 512 20 215040 7 14 21

  • binary Barker to quaternary Golay theorem

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 25 / 33

slide-77
SLIDE 77

Ordered quaternary Golay pairs left to explain

n

# pairs # pairs

n

# pairs # pairs

n

# pairs # pairs

left to left to left to explain explain explain 1 16 8 6656 15 2 64 9 16 106496 3 128 128 10 12288 17 4 512 11 512 512 18 24576 5 512 12 36864 19 6 2048 13 512 20 215040 7 14 21

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 25 / 33

slide-78
SLIDE 78

Ordered quaternary Golay pairs left to explain

n

# pairs # pairs

n

# pairs # pairs

n

# pairs # pairs

left to left to left to explain explain explain 1 16 8 6656 15 2 64 9 16 106496 3 128 128 10 12288 17 4 512 11 512 512 18 24576 5 512 12 36864 19 6 2048 13 512 20 215040 7 14 21

  • ??

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 25 / 33

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SLIDE 79

Constructing a Binary Barker Sequence from a Quaternary Golay Sequence

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 26 / 33

slide-80
SLIDE 80

Goal

  • We have seen that

particular binary Barker

  • f length ≡ 5 (mod 8)

⇒ quaternary Golay

  • f

length ≡ 5 (mod 8)

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 27 / 33

slide-81
SLIDE 81

Goal

  • We have seen that

particular binary Barker

  • f length ≡ 5 (mod 8)

⇒ quaternary Golay

  • f

length ≡ 5 (mod 8)

  • Our (optimistic) objective:

particular binary Barker

  • f length ≡ 5 (mod 8)

⇐ quaternary Golay

  • f

length ≡ 5 (mod 8)

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 27 / 33

slide-82
SLIDE 82

Goal

  • We have seen that

particular binary Barker

  • f length ≡ 5 (mod 8)

⇒ quaternary Golay

  • f

length ≡ 5 (mod 8)

  • Our (optimistic) objective:

particular binary Barker

  • f length ≡ 5 (mod 8)

⇐ quaternary Golay

  • f

length ≡ 5 (mod 8)

  • Since there are no binary Barker sequences of odd length greater than

13, this would prove that there are no more quaternary Golay sequences for these lengths.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 27 / 33

slide-83
SLIDE 83

“Good” sequences

G5,1 = 3 1 G5,2 = 1 2 3

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 28 / 33

slide-84
SLIDE 84

“Good” sequences

G5,1 = 3 1 G5,2 = 1 2 3 G5,1 + G5,2 = 1 2 3

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 28 / 33

slide-85
SLIDE 85

“Good” sequences

G5,1 = 3 1 G5,2 = 1 2 3 G5,1 + G5,2 = 1 2 3 G13,1 = 1 2 3 2 3 1 G13,2 = 1 2 2 2 1 2 3 2 3

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 28 / 33

slide-86
SLIDE 86

“Good” sequences

G5,1 = 3 1 G5,2 = 1 2 3 G5,1 + G5,2 = 1 2 3 G13,1 = 1 2 3 2 3 1 G13,2 = 1 2 2 2 1 2 3 2 3 G13,1 + G13,2 = 1 2 3 1 2 3 1 2 3

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 28 / 33

slide-87
SLIDE 87

“Good sequences” (cont.)

Lemma

If A and B are sequences of length n and A + B = (0 1 2 3 ...), then CB(u) = i−u · CA(u) for all integers 0 < u < n.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 29 / 33

slide-88
SLIDE 88

“Good sequences” (cont.)

Lemma

If A and B are sequences of length n and A + B = (0 1 2 3 ...), then CB(u) = i−u · CA(u) for all integers 0 < u < n. Call a sequence A of length n good if CA(u) = −i−uCA(u) for all integers 0 < u < n.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 29 / 33

slide-89
SLIDE 89

“Good sequences” (cont.)

Lemma

If A and B are sequences of length n and A + B = (0 1 2 3 ...), then CB(u) = i−u · CA(u) for all integers 0 < u < n. Call a sequence A of length n good if CA(u) = −i−uCA(u) for all integers 0 < u < n.

  • A good sequence is necessarily a Golay sequence.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 29 / 33

slide-90
SLIDE 90

“Good sequences” (cont.)

Lemma

If A and B are sequences of length n and A + B = (0 1 2 3 ...), then CB(u) = i−u · CA(u) for all integers 0 < u < n. Call a sequence A of length n good if CA(u) = −i−uCA(u) for all integers 0 < u < n.

  • A good sequence is necessarily a Golay sequence.
  • G5,1, G5,2, G13,1, and G13,2 are all good.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 29 / 33

slide-91
SLIDE 91

“Good sequences” (cont.)

Lemma

If A and B are sequences of length n and A + B = (0 1 2 3 ...), then CB(u) = i−u · CA(u) for all integers 0 < u < n. Call a sequence A of length n good if CA(u) = −i−uCA(u) for all integers 0 < u < n.

  • A good sequence is necessarily a Golay sequence.
  • G5,1, G5,2, G13,1, and G13,2 are all good.
  • Output sequences of Barker-to-Golay theorem are good.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 29 / 33

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SLIDE 92

A partial Barker-to-Golay converse

Theorem

Let A = (a0, ..., an−1) be a good sequence of length n = 8m + 5. Assume that (1) a2u−1 + a2u+1 ≡ 1 (mod 2), for all 1 ≤ 2u − 1 ≤ n−7

2 , and

(2) a4u ≡ 0 (mod 2), for all 4 ≤ 4u ≤ n−5

2 .

Then there exists a binary Barker sequence of length n, and so m ∈ {0, 1}.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 30 / 33

slide-93
SLIDE 93

A partial Barker-to-Golay converse

Theorem

Let A = (a0, ..., an−1) be a good sequence of length n = 8m + 5. Assume that (1) a2u−1 + a2u+1 ≡ 1 (mod 2), for all 1 ≤ 2u − 1 ≤ n−7

2 , and

(2) a4u ≡ 0 (mod 2), for all 4 ≤ 4u ≤ n−5

2 .

Then there exists a binary Barker sequence of length n, and so m ∈ {0, 1}. Proof: About 20 pages of lemmas.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 30 / 33

slide-94
SLIDE 94

Summary and Open Problems

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 31 / 33

slide-95
SLIDE 95

Summary and open problems

Summary of results:

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 32 / 33

slide-96
SLIDE 96

Summary and open problems

Summary of results:

  • Classified and explained (almost) all ordered quaternary Golay pairs of

length less than 22.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 32 / 33

slide-97
SLIDE 97

Summary and open problems

Summary of results:

  • Classified and explained (almost) all ordered quaternary Golay pairs of

length less than 22.

  • Found a general construction of a quaternary Golay pair from a

particular type of binary Barker sequence.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 32 / 33

slide-98
SLIDE 98

Summary and open problems

Summary of results:

  • Classified and explained (almost) all ordered quaternary Golay pairs of

length less than 22.

  • Found a general construction of a quaternary Golay pair from a

particular type of binary Barker sequence.

  • Established a partial converse to the Barker-to-Golay construction.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 32 / 33

slide-99
SLIDE 99

Summary and open problems

Summary of results:

  • Classified and explained (almost) all ordered quaternary Golay pairs of

length less than 22.

  • Found a general construction of a quaternary Golay pair from a

particular type of binary Barker sequence.

  • Established a partial converse to the Barker-to-Golay construction.

Open problems:

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 32 / 33

slide-100
SLIDE 100

Summary and open problems

Summary of results:

  • Classified and explained (almost) all ordered quaternary Golay pairs of

length less than 22.

  • Found a general construction of a quaternary Golay pair from a

particular type of binary Barker sequence.

  • Established a partial converse to the Barker-to-Golay construction.

Open problems:

  • Explain the existence of the length 3 and 11 quaternary Golay pairs.

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 32 / 33

slide-101
SLIDE 101

Summary and open problems

Summary of results:

  • Classified and explained (almost) all ordered quaternary Golay pairs of

length less than 22.

  • Found a general construction of a quaternary Golay pair from a

particular type of binary Barker sequence.

  • Established a partial converse to the Barker-to-Golay construction.

Open problems:

  • Explain the existence of the length 3 and 11 quaternary Golay pairs.
  • Are there any quaternary Golay sequences of odd length greater than

13?

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 32 / 33

slide-102
SLIDE 102

Summary and open problems

Summary of results:

  • Classified and explained (almost) all ordered quaternary Golay pairs of

length less than 22.

  • Found a general construction of a quaternary Golay pair from a

particular type of binary Barker sequence.

  • Established a partial converse to the Barker-to-Golay construction.

Open problems:

  • Explain the existence of the length 3 and 11 quaternary Golay pairs.
  • Are there any quaternary Golay sequences of odd length greater than

13?

  • Odd length ⇒ good?

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 32 / 33

slide-103
SLIDE 103

Summary and open problems

Summary of results:

  • Classified and explained (almost) all ordered quaternary Golay pairs of

length less than 22.

  • Found a general construction of a quaternary Golay pair from a

particular type of binary Barker sequence.

  • Established a partial converse to the Barker-to-Golay construction.

Open problems:

  • Explain the existence of the length 3 and 11 quaternary Golay pairs.
  • Are there any quaternary Golay sequences of odd length greater than

13?

  • Odd length ⇒ good?
  • Can we overcome conditions (1) and (2) of Barker-to-Golay converse?

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 32 / 33

slide-104
SLIDE 104

Summary and open problems

Summary of results:

  • Classified and explained (almost) all ordered quaternary Golay pairs of

length less than 22.

  • Found a general construction of a quaternary Golay pair from a

particular type of binary Barker sequence.

  • Established a partial converse to the Barker-to-Golay construction.

Open problems:

  • Explain the existence of the length 3 and 11 quaternary Golay pairs.
  • Are there any quaternary Golay sequences of odd length greater than

13?

  • Odd length ⇒ good?
  • Can we overcome conditions (1) and (2) of Barker-to-Golay converse?
  • Alternative approach?

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 32 / 33

slide-105
SLIDE 105

Thanks for listening!

Quaternary Golay Sequence Pairs : Richard Gibson (SFU) 33 / 33