On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a...
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Costas ArraysCross-correlation(Partial) Solution
Summary
On the roots of a polynomial connected withGolomb Costas Arrays
John Sheekey
Claude Shannon InstituteSchool of Mathematical Science
University College Dublin
23 July 2010 / Fields Institute-Carleton Finite FieldsWorkshop
John Sheekey On the roots of a polynomial connected with Costas Arrays
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Costas ArraysCross-correlation(Partial) Solution
Summary
Outline
1 Costas Arrays
2 Cross-correlation
3 (Partial) Solution
John Sheekey On the roots of a polynomial connected with Costas Arrays
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Costas ArraysCross-correlation(Partial) Solution
Summary
DefinitionA Costas Array C (of order n) is an n × n grid containing n dotssuch that
Each row and each column contains precisely one dot(permutation matrix)All displacement vectors (i.e. vector between two dots) aredistinct
In other words, the autocorrelation function of C is always either0 or 1.
John Sheekey On the roots of a polynomial connected with Costas Arrays
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Costas ArraysCross-correlation(Partial) Solution
Summary
DefinitionA Costas Array C (of order n) is an n × n grid containing n dotssuch that
Each row and each column contains precisely one dot(permutation matrix)All displacement vectors (i.e. vector between two dots) aredistinct
In other words, the autocorrelation function of C is always either0 or 1.
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 5: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/5.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
DefinitionA Costas Array C (of order n) is an n × n grid containing n dotssuch that
Each row and each column contains precisely one dot(permutation matrix)All displacement vectors (i.e. vector between two dots) aredistinct
In other words, the autocorrelation function of C is always either0 or 1.
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 6: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/6.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Construction
Applications in radar and sonarThe number of Costas Arrays of a given order is notknown. In fact, the existence of Costas Arrays for all n is anopen problem.However, there are some constructions.
John Sheekey On the roots of a polynomial connected with Costas Arrays
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Costas ArraysCross-correlation(Partial) Solution
Summary
Construction
Applications in radar and sonarThe number of Costas Arrays of a given order is notknown. In fact, the existence of Costas Arrays for all n is anopen problem.However, there are some constructions.
John Sheekey On the roots of a polynomial connected with Costas Arrays
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Costas ArraysCross-correlation(Partial) Solution
Summary
Construction
Applications in radar and sonarThe number of Costas Arrays of a given order is notknown. In fact, the existence of Costas Arrays for all n is anopen problem.However, there are some constructions.
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 9: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/9.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Definition (Welch Array)Let α be a primitive element of Fp, p a prime. Define apermutation π on {1..p − 1} by
π(i) = αi
Then π is a Costas permutation
Definition (Golomb Array)Let α and β be primitive elements of Fq, q a power of a prime.Define a permutation π on {1..q − 2} by
αi + βπ(i) = 1
Then π is a Costas permutation. Denote this by Gα,β
John Sheekey On the roots of a polynomial connected with Costas Arrays
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Costas ArraysCross-correlation(Partial) Solution
Summary
Definition (Welch Array)Let α be a primitive element of Fp, p a prime. Define apermutation π on {1..p − 1} by
π(i) = αi
Then π is a Costas permutation
Definition (Golomb Array)Let α and β be primitive elements of Fq, q a power of a prime.Define a permutation π on {1..q − 2} by
αi + βπ(i) = 1
Then π is a Costas permutation. Denote this by Gα,β
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 11: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/11.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Suppose we had two Golomb arrays of the same order, Gα,β
and Gαr ,βs , where (r ,q − 1) = (s,q − 1) = 1. Then themaximum cross-correlation between the two arrays can beshown to equal the number of roots of the polynomial
Fr ,s(z) := zr + (1− z)s − 1
in Fq.
Conjecture (Rickard)
Fr ,s has at most q+12 roots in Fq
John Sheekey On the roots of a polynomial connected with Costas Arrays
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Costas ArraysCross-correlation(Partial) Solution
Summary
Suppose we had two Golomb arrays of the same order, Gα,β
and Gαr ,βs , where (r ,q − 1) = (s,q − 1) = 1. Then themaximum cross-correlation between the two arrays can beshown to equal the number of roots of the polynomial
Fr ,s(z) := zr + (1− z)s − 1
in Fq.
Conjecture (Rickard)
Fr ,s has at most q+12 roots in Fq
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 13: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/13.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Suppose we had two Golomb arrays of the same order, Gα,β
and Gαr ,βs , where (r ,q − 1) = (s,q − 1) = 1. Then themaximum cross-correlation between the two arrays can beshown to equal the number of roots of the polynomial
Fr ,s(z) := zr + (1− z)s − 1
in Fq.
Conjecture (Rickard)
Fr ,s has at most q+12 roots in Fq
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 14: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/14.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Suppose we had two Golomb arrays of the same order, Gα,β
and Gαr ,βs , where (r ,q − 1) = (s,q − 1) = 1. Then themaximum cross-correlation between the two arrays can beshown to equal the number of roots of the polynomial
Fr ,s(z) := zr + (1− z)s − 1
in Fq.
Conjecture (Rickard)
Fr ,s has at most q+12 roots in Fq
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 15: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/15.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
We consider the case r = s, r odd, and denote by Fr .
0 and 1 are roots for all r .
Fr (z) = Fr (1− z) = −zr Fr (1z
)
If α is a root, then 1− α is a rootIf α 6= 0 is a root, then 1
α is a rootSo there is an action by S3 on the roots of the polynomialThis polynomial also arises in the cross-correlation ofm-sequences, and in the study of APN functionsIt is related to Cauchy-Mirimanoff polynomials
John Sheekey On the roots of a polynomial connected with Costas Arrays
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Costas ArraysCross-correlation(Partial) Solution
Summary
We consider the case r = s, r odd, and denote by Fr .
0 and 1 are roots for all r .
Fr (z) = Fr (1− z) = −zr Fr (1z
)
If α is a root, then 1− α is a rootIf α 6= 0 is a root, then 1
α is a rootSo there is an action by S3 on the roots of the polynomialThis polynomial also arises in the cross-correlation ofm-sequences, and in the study of APN functionsIt is related to Cauchy-Mirimanoff polynomials
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 17: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/17.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
We consider the case r = s, r odd, and denote by Fr .
0 and 1 are roots for all r .
Fr (z) = Fr (1− z) = −zr Fr (1z
)
If α is a root, then 1− α is a rootIf α 6= 0 is a root, then 1
α is a rootSo there is an action by S3 on the roots of the polynomialThis polynomial also arises in the cross-correlation ofm-sequences, and in the study of APN functionsIt is related to Cauchy-Mirimanoff polynomials
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 18: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/18.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
We consider the case r = s, r odd, and denote by Fr .
0 and 1 are roots for all r .
Fr (z) = Fr (1− z) = −zr Fr (1z
)
If α is a root, then 1− α is a rootIf α 6= 0 is a root, then 1
α is a rootSo there is an action by S3 on the roots of the polynomialThis polynomial also arises in the cross-correlation ofm-sequences, and in the study of APN functionsIt is related to Cauchy-Mirimanoff polynomials
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 19: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/19.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
We consider the case r = s, r odd, and denote by Fr .
0 and 1 are roots for all r .
Fr (z) = Fr (1− z) = −zr Fr (1z
)
If α is a root, then 1− α is a rootIf α 6= 0 is a root, then 1
α is a rootSo there is an action by S3 on the roots of the polynomialThis polynomial also arises in the cross-correlation ofm-sequences, and in the study of APN functionsIt is related to Cauchy-Mirimanoff polynomials
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 20: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/20.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
We consider the case r = s, r odd, and denote by Fr .
0 and 1 are roots for all r .
Fr (z) = Fr (1− z) = −zr Fr (1z
)
If α is a root, then 1− α is a rootIf α 6= 0 is a root, then 1
α is a rootSo there is an action by S3 on the roots of the polynomialThis polynomial also arises in the cross-correlation ofm-sequences, and in the study of APN functionsIt is related to Cauchy-Mirimanoff polynomials
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 21: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/21.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
We consider the case r = s, r odd, and denote by Fr .
0 and 1 are roots for all r .
Fr (z) = Fr (1− z) = −zr Fr (1z
)
If α is a root, then 1− α is a rootIf α 6= 0 is a root, then 1
α is a rootSo there is an action by S3 on the roots of the polynomialThis polynomial also arises in the cross-correlation ofm-sequences, and in the study of APN functionsIt is related to Cauchy-Mirimanoff polynomials
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 22: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/22.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
LemmaLet r be odd. Let S denote the set of non-zero roots of Fr overFq. Suppose x and y are in S, with y 6= 1. Then
xy∈ S ⇔ 1− x
1− y∈ S
Proof.x and y are roots of Fr , so
x r + (1− x)r = 1y r + (1− y)r = 1
⇒ x r − y r = (1− y)r − (1− x)r
John Sheekey On the roots of a polynomial connected with Costas Arrays
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Costas ArraysCross-correlation(Partial) Solution
Summary
LemmaLet r be odd. Let S denote the set of non-zero roots of Fr overFq. Suppose x and y are in S, with y 6= 1. Then
xy∈ S ⇔ 1− x
1− y∈ S
Proof.x and y are roots of Fr , so
x r + (1− x)r = 1y r + (1− y)r = 1
⇒ x r − y r = (1− y)r − (1− x)r
John Sheekey On the roots of a polynomial connected with Costas Arrays
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Costas ArraysCross-correlation(Partial) Solution
Summary
LemmaLet r be odd. Let S denote the set of non-zero roots of Fr overFq. Suppose x and y are in S, with y 6= 1. Then
xy∈ S ⇔ 1− x
1− y∈ S
Proof.x and y are roots of Fr , so
x r + (1− x)r = 1y r + (1− y)r = 1
⇒ x r − y r = (1− y)r − (1− x)r
John Sheekey On the roots of a polynomial connected with Costas Arrays
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Costas ArraysCross-correlation(Partial) Solution
Summary
LemmaLet r be odd. Let S denote the set of non-zero roots of Fr overFq. Suppose x and y are in S, with y 6= 1. Then
xy∈ S ⇔ 1− x
1− y∈ S
Proof.x and y are roots of Fr , so
x r + (1− x)r = 1y r + (1− y)r = 1
⇒ x r − y r = (1− y)r − (1− x)r
John Sheekey On the roots of a polynomial connected with Costas Arrays
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Costas ArraysCross-correlation(Partial) Solution
Summary
LemmaLet r be odd. Let S denote the set of non-zero roots of Fr overFq. Suppose x and y are in S, with y 6= 1. Then
xy∈ S ⇔ 1− x
1− y∈ S
Proof.x and y are roots of Fr , so
x r + (1− x)r = 1y r + (1− y)r = 1
⇒ x r − y r = (1− y)r − (1− x)r
John Sheekey On the roots of a polynomial connected with Costas Arrays
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Costas ArraysCross-correlation(Partial) Solution
Summary
Proof(contd.)
Then xy is a root
⇔ ( xy )r + (1− x
y )r = 1⇔ x r + (y − x)r = y r
⇔ x r − y r = (x − y)r
⇔ (1− y)r − (1− x)r = (x − y)r
⇔ (1− x)r + (x − y)r = (1− y)r
⇔ (1−x1−y )r + ( x−y
1−y )r = 1
⇔ 1−x1−y is a root of Fr
John Sheekey On the roots of a polynomial connected with Costas Arrays
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Costas ArraysCross-correlation(Partial) Solution
Summary
Proof(contd.)
Then xy is a root
⇔ ( xy )r + (1− x
y )r = 1⇔ x r + (y − x)r = y r
⇔ x r − y r = (x − y)r
⇔ (1− y)r − (1− x)r = (x − y)r
⇔ (1− x)r + (x − y)r = (1− y)r
⇔ (1−x1−y )r + ( x−y
1−y )r = 1
⇔ 1−x1−y is a root of Fr
John Sheekey On the roots of a polynomial connected with Costas Arrays
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Costas ArraysCross-correlation(Partial) Solution
Summary
Proof(contd.)
Then xy is a root
⇔ ( xy )r + (1− x
y )r = 1⇔ x r + (y − x)r = y r
⇔ x r − y r = (x − y)r
⇔ (1− y)r − (1− x)r = (x − y)r
⇔ (1− x)r + (x − y)r = (1− y)r
⇔ (1−x1−y )r + ( x−y
1−y )r = 1
⇔ 1−x1−y is a root of Fr
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 30: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/30.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Proof(contd.)
Then xy is a root
⇔ ( xy )r + (1− x
y )r = 1⇔ x r + (y − x)r = y r
⇔ x r − y r = (x − y)r
⇔ (1− y)r − (1− x)r = (x − y)r
⇔ (1− x)r + (x − y)r = (1− y)r
⇔ (1−x1−y )r + ( x−y
1−y )r = 1
⇔ 1−x1−y is a root of Fr
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 31: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/31.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Proof(contd.)
Then xy is a root
⇔ ( xy )r + (1− x
y )r = 1⇔ x r + (y − x)r = y r
⇔ x r − y r = (x − y)r
⇔ (1− y)r − (1− x)r = (x − y)r
⇔ (1− x)r + (x − y)r = (1− y)r
⇔ (1−x1−y )r + ( x−y
1−y )r = 1
⇔ 1−x1−y is a root of Fr
John Sheekey On the roots of a polynomial connected with Costas Arrays
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Costas ArraysCross-correlation(Partial) Solution
Summary
Proof(contd.)
Then xy is a root
⇔ ( xy )r + (1− x
y )r = 1⇔ x r + (y − x)r = y r
⇔ x r − y r = (x − y)r
⇔ (1− y)r − (1− x)r = (x − y)r
⇔ (1− x)r + (x − y)r = (1− y)r
⇔ (1−x1−y )r + ( x−y
1−y )r = 1
⇔ 1−x1−y is a root of Fr
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 33: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/33.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Proof(contd.)
Then xy is a root
⇔ ( xy )r + (1− x
y )r = 1⇔ x r + (y − x)r = y r
⇔ x r − y r = (x − y)r
⇔ (1− y)r − (1− x)r = (x − y)r
⇔ (1− x)r + (x − y)r = (1− y)r
⇔ (1−x1−y )r + ( x−y
1−y )r = 1
⇔ 1−x1−y is a root of Fr
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 34: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/34.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Proof(contd.)
Then xy is a root
⇔ ( xy )r + (1− x
y )r = 1⇔ x r + (y − x)r = y r
⇔ x r − y r = (x − y)r
⇔ (1− y)r − (1− x)r = (x − y)r
⇔ (1− x)r + (x − y)r = (1− y)r
⇔ (1−x1−y )r + ( x−y
1−y )r = 1
⇔ 1−x1−y is a root of Fr
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 35: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/35.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Proof(contd.)
Then xy is a root
⇔ ( xy )r + (1− x
y )r = 1⇔ x r + (y − x)r = y r
⇔ x r − y r = (x − y)r
⇔ (1− y)r − (1− x)r = (x − y)r
⇔ (1− x)r + (x − y)r = (1− y)r
⇔ (1−x1−y )r + ( x−y
1−y )r = 1
⇔ 1−x1−y is a root of Fr
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 36: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/36.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Applying this result to 1x and 1
y , we also have
CorollarySuppose x and y are in S, with y 6= 1. Then
xy∈ S ⇔ y
x(1− x1− y
) ∈ S
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 37: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/37.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Applying this result to 1x and 1
y , we also have
CorollarySuppose x and y are in S, with y 6= 1. Then
xy∈ S ⇔ y
x(1− x1− y
) ∈ S
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 38: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/38.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Suppose now that c is any non-root of Fr . Consider the set
1c
S = {x | Fr (cx) = 0}
Let x ∈ S ∩ 1c S, i.e. x and cx are both roots of Fr . Then by the
previous lemma,1− x1− cx
andc(
1− x1− cx
)
are both non-roots of Fr (as c = cxx is not a root). Hence for
every element x of S ∩ 1c S, there is an element 1−x
1−cx which isnot in S ∪ 1
c S.
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 39: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/39.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Suppose now that c is any non-root of Fr . Consider the set
1c
S = {x | Fr (cx) = 0}
Let x ∈ S ∩ 1c S, i.e. x and cx are both roots of Fr . Then by the
previous lemma,1− x1− cx
andc(
1− x1− cx
)
are both non-roots of Fr (as c = cxx is not a root). Hence for
every element x of S ∩ 1c S, there is an element 1−x
1−cx which isnot in S ∪ 1
c S.
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 40: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/40.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Suppose now that c is any non-root of Fr . Consider the set
1c
S = {x | Fr (cx) = 0}
Let x ∈ S ∩ 1c S, i.e. x and cx are both roots of Fr . Then by the
previous lemma,1− x1− cx
andc(
1− x1− cx
)
are both non-roots of Fr (as c = cxx is not a root). Hence for
every element x of S ∩ 1c S, there is an element 1−x
1−cx which isnot in S ∪ 1
c S.
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 41: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/41.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Suppose now that c is any non-root of Fr . Consider the set
1c
S = {x | Fr (cx) = 0}
Let x ∈ S ∩ 1c S, i.e. x and cx are both roots of Fr . Then by the
previous lemma,1− x1− cx
andc(
1− x1− cx
)
are both non-roots of Fr (as c = cxx is not a root). Hence for
every element x of S ∩ 1c S, there is an element 1−x
1−cx which isnot in S ∪ 1
c S.
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 42: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/42.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Suppose now that c is any non-root of Fr . Consider the set
1c
S = {x | Fr (cx) = 0}
Let x ∈ S ∩ 1c S, i.e. x and cx are both roots of Fr . Then by the
previous lemma,1− x1− cx
andc(
1− x1− cx
)
are both non-roots of Fr (as c = cxx is not a root). Hence for
every element x of S ∩ 1c S, there is an element 1−x
1−cx which isnot in S ∪ 1
c S.
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 43: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/43.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Suppose now that c is any non-root of Fr . Consider the set
1c
S = {x | Fr (cx) = 0}
Let x ∈ S ∩ 1c S, i.e. x and cx are both roots of Fr . Then by the
previous lemma,1− x1− cx
andc(
1− x1− cx
)
are both non-roots of Fr (as c = cxx is not a root). Hence for
every element x of S ∩ 1c S, there is an element 1−x
1−cx which isnot in S ∪ 1
c S.
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 44: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/44.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Suppose now that c is any non-root of Fr . Consider the set
1c
S = {x | Fr (cx) = 0}
Let x ∈ S ∩ 1c S, i.e. x and cx are both roots of Fr . Then by the
previous lemma,1− x1− cx
andc(
1− x1− cx
)
are both non-roots of Fr (as c = cxx is not a root). Hence for
every element x of S ∩ 1c S, there is an element 1−x
1−cx which isnot in S ∪ 1
c S.
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 45: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/45.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
So if we setU = { 1− x
1− cx| x ∈ S ∩ 1
cS}
we have that |U| = |S ∩ 1c S|, and hence
|U ∪ S ∪ 1c
S| = 2|S| ≤ q − 1
proving the result:
TheoremIf r is odd and p − 1 does not divide r − 1, then the polynomial
zr + (1− z)r − 1
has at most q+12 roots in Fq.
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 46: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/46.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
So if we setU = { 1− x
1− cx| x ∈ S ∩ 1
cS}
we have that |U| = |S ∩ 1c S|, and hence
|U ∪ S ∪ 1c
S| = 2|S| ≤ q − 1
proving the result:
TheoremIf r is odd and p − 1 does not divide r − 1, then the polynomial
zr + (1− z)r − 1
has at most q+12 roots in Fq.
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 47: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/47.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
So if we setU = { 1− x
1− cx| x ∈ S ∩ 1
cS}
we have that |U| = |S ∩ 1c S|, and hence
|U ∪ S ∪ 1c
S| = 2|S| ≤ q − 1
proving the result:
TheoremIf r is odd and p − 1 does not divide r − 1, then the polynomial
zr + (1− z)r − 1
has at most q+12 roots in Fq.
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 48: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/48.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
So if we setU = { 1− x
1− cx| x ∈ S ∩ 1
cS}
we have that |U| = |S ∩ 1c S|, and hence
|U ∪ S ∪ 1c
S| = 2|S| ≤ q − 1
proving the result:
TheoremIf r is odd and p − 1 does not divide r − 1, then the polynomial
zr + (1− z)r − 1
has at most q+12 roots in Fq.
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 49: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/49.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Summary
We have proved Rickard’s Conjecture for the case r = s
Future workr 6= s?Exact number of roots?Fr irreducible over Z[z]?
John Sheekey On the roots of a polynomial connected with Costas Arrays
![Page 50: On the roots of a polynomial connected with Golomb Costas Arrays · John Sheekey On the roots of a polynomial connected with Costas Arrays. Costas Arrays Cross-correlation (Partial)](https://reader034.fdocuments.in/reader034/viewer/2022042812/5fb196193e9172405840f3cb/html5/thumbnails/50.jpg)
Costas ArraysCross-correlation(Partial) Solution
Summary
Summary
We have proved Rickard’s Conjecture for the case r = s
Future workr 6= s?Exact number of roots?Fr irreducible over Z[z]?
John Sheekey On the roots of a polynomial connected with Costas Arrays