Contents Introduction Related problems Constructions –Welch construction –Lempel construction –Golomb construction Special properties –Periodicity –Nonattacking

Contents• Introduction• Related problems• Constructions

–Welch construction–Lempel construction–Golomb construction

• Special properties–Periodicity–Nonattacking queens–Shearing–Honeycomb arrays–Nonattacking kings

• Unsolved problems

Introduction

We want to find 2D patterns of ones (dots) and zeros (blanks) for which the planar autocorrelation function has minimum out-of-phase values.

Introduction

We want to find 2D patterns of ones (dots) and zeros (blanks) for which the planar autocorrelation function has minimum out-of-phase values.

These “minimum agreement” patterns may be viewed as a generalization of the 1D “ruler problems”.

For example minimizes in a array with four dots.

1 nn

Related problemsWe consider patterns of dots in rectangular grid

under the following requirements:1. For horizontal and vertical noncyclic shifting, the

shifted pattern will overlap with the original one in at most one dot

ProblemLet denote the maximal number of dots in an array.Known values of :

For every there exists a construction, for , such that (it is not known if holds for all ).

On the other hand, it is proven that there exists a constant such that (Erdos-Turan).

( )g n n n()g

(1) 1, (2) 3, (3) 5, (4) 6, (5) 8g g g g g

N n N( )g n n n

2

3( )g n n kn

k( )g n n

Related problemsWe consider patterns of dots in rectangular grid



2. The patterns will be of size and will have one dot in every row and column

n n

ProblemThese patterns exist for infinitely many values of , nevertheless, there is no construction for general value of .

There are (permutations).

We denote to be the sequence of

It seems very probable that and therefore we will not rely on random constructions of arrays.

nn

!n np

#( )

!

valid n n array

n

1 2 3 4 5 6 7p p p p p p p

0n np

RequirementsWe consider patterns of dots in rectangular grid



3. The patterns will be of size and will have one dot in every column (rows are not restricted)

n m

SonarIn the sonar application our array is a sequence of distinct frequencies in consecutive time slots .

Vertical shift happens due to the Doppler effect (velocity) and horizontal shift happens due to the time elapsed (range).

In this case our objective is to maximize the number columns for a given .

( )n ( )m

n




( )n ( )m

n

4 8




• • Let be maximal m for which the out-of-phase

agreement is at most dots.

( )n ( )m

n

2m n

km

k(2 1) 1km k n

Sonar

for

• • Let be maximal m for which the out-of-phase

agreement is at most dots.

2m n

km

k(2 1) 1km k n

2k

3 10

Related problemWe consider patterns of dots in rectangular grid

under the following requirements:3. The patterns will be of size and will have one dot

in every column (rows are not restricted)

4. For only horizontal noncyclic shifting, the shifted pattern will overlap with the original one in at most one dot

n m

RadarRadar application might not require Doppler measurement and will not care for vertical shifts.

We would still try to maximize the number columns for a given .• “max columns”

n

2n 3n

Difference triangle algorithmThis is an algorithm to check the validity of a

array with one dot per column.

This is the corresponding triangle to the array .

To check the validity of the array, make sure that the rows

in the triangle, have no repeated signed differences.

n m

Connection to 1DA connection between 1D “ruler problems”, and some of the

2D min-agreement patterns can be established through the

following:

Another problemOne more related problem is to find a pattern with the minimal number of dots in a rectangle, such that no additional dot can be placed without causing a repeat pair.

n m

ConstructionsThe objective:For each construct a permutation matrix such that, difference vectors are all distinct as vectors.

Such matrices are called Costas Arrays.

n n n

2

n

Welch constructionTheorem 1:Let be a primitive root modulo the prime .

Then the permutation matrix with

iff ,

g p

( 1) ( 1)p p 1ija (mod )ij g p , {1,..., 1}i j p



iff ,

Proof:Consider the two pairs

where , and assume equality of the difference vectors , and then

Hence this pattern is a Costas Array

g p


{( , ), ( , )}, {( , ), ( , )}i i k l l ki g i k g l g l k g {1,..., 2}k p

( , ) ( , )i k i l k lk g g k g g ( 1) ( 1)i k l k i lg g g g g g i l



iff ,

Example:For

the Costas Array is

g p


7, 3p g

Cont’Lemma 1:If an Costas Array has 1 in any of its four corners, the corresponding row and column can be removed to obtain an

Costas Array.

Proof:Any violation in the reduced pattern would have been a violation I the original pattern.

n n

( 1) ( 1)n n

Cont’Corollary 1.1:We can obtain a Costas Array, from the Welch construction.

Proof:Since , the original array of degree

has a 1 in which is a corner, and so by Lemma 1,

the array can be reduced to degree .

( 2) ( 2)p p

1 1(mod )pg p 1p ( 1,1)p

1p

Cont’Corollary 1.2:If 2 is a primitive root modulo then,

we can obtain a Costas Array, from the Welch construction.

Proof:After removing the 1 at position , the 1 at position

becomes a corner, and can also be removed by Lemma 1.

( 3) ( 3)p p

( 1,1)p

p

(1,2)

Cont’Corollary 1.3:Every cyclic permutation of the rows of a Costas Array in the

Welch construction, is again a Costas Array.

Proof:Let be a primitive root modulo and let be any fixed positive integer. Then the permutation matrix with

iff , is a Costas Array of degree and the successive values of give the successive cyclic permutation of the rows in the Welch construction.

Note:Costas Array property is preserved under the group of dihedral symmetries of the square.

( 1) ( 1)p p

1p

g p c1ija

(mod )i cj g pc

4Dn n

Lempel constructionTheorem 2:Let be a primitive element in the field , for

then the symmetric permutation matrix with

iff , , is a Costas Array.

( )GF q

( 2) ( 2)q q 1ija 1i j , {1,..., 2}i j q

2kq p

Lempel constructionTheorem 2:Let be a primitive element in the field , for

then the symmetric permutation matrix with


Proof:If we may write . Consider the two pairs

and assume equality of the difference vectors

and then

( )GF q

( 2) ( 2)q q 1ija 1i j , {1,..., 2}i j q

2kq p

1i j log (1 )ij {( , log (1 )), ( , log (1 ))},i i ki i k

{( , log (1 )), ( , log (1 ))}l l kl l k

1 1 1 1log ( ) log ( )

1 1 1 1

i k l k i k l k

i l i l

1 1l i k i l k i l k i l k

Lempel construction

and since , the above requires .

Hence this pattern is a Costas Array.

( 1) ( 1)l k l i k i l k i k 1 0k i l

Lempel construction

and since , the above requires .

Hence this pattern is a Costas Array.

Example:Let and let be a root of the polynomial

.

The Costas Array is

( 1) ( 1)l k l i k i l k i k 1 0k i l

32 8q 3 1 0x x 6 6

Cont’Corollary 2.1:If 2 is a primitive root of modulo , then a symmetric

Costas Array can be constructed.

Proof:Taking and , we get that

which is a corner, and so by applying Lemma 1, we reduce the pattern to degree

p

( 3) ( 3)p p

q p 2 12 1(mod )p p 2 2

2, 22 2 1(mod ) 1p pp pp a

3.p

Golomb constructionTheorem 3:Let be primitive elements in the field , for

then the permutation matrix with


(generalization of Theorem 2)

, ( )GF q

( 2) ( 2)q q 1ija 1i j , {1,..., 2}i j q

2kq p

Golomb constructionTheorem 3:Let be primitive elements in the field , for

then the permutation matrix with


(generalization of Theorem 2)

Proof:If we may write . The rest is exactly as in the proof of Theorem 2.

, ( )GF q

( 2) ( 2)q q 1ija 1i j , {1,..., 2}i j q

2kq p

1i j log (1 )ij

Cont’Example:Let and let be a root of the polynomial

, and let .

The Costas Array is

32 8q 3 3 1 0x x

6 6


, and let .

The Costas Array is

32 8q 3 3 1 0x x

6 6


, and let .

The Costas Array is

Additional Corollaries:• Corollary 2.2 (Taylor): if in satisfies then by removing we get Costas Array of degree .

• Corollary 3.1: if then by removing we get Costas Array of degree .

• Corollary 3.2: if and then and so by removing we get Costas Array of degree .

32 8q 3 3 1 0x x

6 6

1 2 1 ( )GF q(1,2), (2,1) 4q

1 1 1 (1,1)3q 1 1 1 2kq 2 2 1

(1,1), (2, 2) 4q

Cont’• Corollary 3.3: if and by the arithmetic of exponents and thus by removing , we will get a pattern of degree .

• Corollary 3.4: if and then necessarily

and thus by removing , we will get a pattern of degree .

1 1 1 2 1 1 1 2q (1,1), (2, 2)q

4q 1 1 1 2 1 1

1 2 1 (1,1), (2, 2), ( 2,2)q q 5q

Special properties• PeriodicityRepeating Costas Array in both directions over the entire plain, gives a doubly periodic checkerboard pattern.

It is proven that for all , there does not exist a doubly periodic pattern.

Repeating the degree pattern, which we obtain from the Welch construction, will get us singly periodic pattern.

2 2

2n

1p ( 1)p

Special properties• Nonattacking QueensFor there is no known Costas Array consisting of nonattacking queens.

Regarding semi-Queens, we have an infinite supply of patterns, from the Lempel construction, which are valid due to the symmetry of the Lempel construction.

1n


Regarding semi-Queens, we have an infinite supply of patterns, from the Lempel construction, which are valid due to the symmetry of the Lempel construction.

Semi-Queen – attacks its row, column, and only the diagonal parallel to the main diagonal.

1n


Regarding semi-Queens, we have an infinite supply of patterns, from the Lempel construction, which are valid due to the symmetry of the construction.– If is a power of an odd prime there will be exactly one dot on the main diagonal, and if is a power of 2, there will be none.

1n

qq

Special properties• ShearingDistinctness of differences is preserved while applying non-singular linear transformation.

For example: using

linear transformation on

the Lempel construction

with will in

fact produce a

rotation of itself.

1 1

0 1

11, 3q 90

Special Properties• ShearingDistinctness of differences is preserved while applying non-singular linear transformation.

For example: using

linear transformation on

the Lempel construction

with will in

fact produce a

rotation of itself.

1 1

0 1

11, 3q 90

Shearing cont’• Only few Costas Arrays are shearable by into another Costas Array. For this to happen, the array must have one dot in each of consecutive lines parallel to the main diagonal – those become the columns after the shearing. The rows remain rows, and the columns become lines orthogonal to the main diagonal.• Almost all shearable arrays go through a cycle of four different patterns via shearing alternately by and

1 1

0 1

n

1 1

0 1

1 0

1 1

Shearing cont’The following array goes through a cycle of twelve patterns:

Special properties• Honeycomb ArraysShear-compression by will convert the square cells into hexagonal cells.

When dealing with Costas Arrays with nonattacking semi-queens we can delete the unoccupied diagonal lines, and apply shear-compression to get a ”honeycomb array”.

The semi-queens become (still) nonattacking bee-Rooks.

3 1

2 20 1

1n n

Honeycomb Arrays

Honeycomb ArraysSome definitions:

Bee-Dukes – is a piece which can move to any one of the six adjacent cells (on hexagonal boards).

The distance between two cells in the hexagonal Lee metric – is the minimal number of bee-Duke moves needed to go from one cell to another.

Lee sphere of radius – consists of a center cell, together with all the cells at distance .

Note:

All known honeycomb arrays with nonattacking bee-Rooks are in fact a Lee sphere, but it is not yet proven to always be the case.

rr

n

Honeycomb ArraysSome related problems:• The cuban primes of Cunningham show up when we count the number of cells on a Lee sphere of radius . This number is always of the form , and is often a prime.• The zero sum arrays of Bennett and Potts arrive at the problem of counting , which is the number of configurations of nonattacking bee-Rooks on a honeycomb which is a Lee sphere of radius .

Let be the number of configurations inequivalent under the dihedral group of symmetries of the hexagon.

r3 3( 1)r r

( )N r2 1n r

r( )r

Honeycomb ArraysSome related problems:• The cuban primes of Cunningham show up when we count the number of cells on a Lee sphere of radius . This number is always of the form , and is often a prime.• The zero sum arrays of Bennett and Potts arrive at the problem of counting , which is the number of configurations of nonattacking bee-Rooks on a honeycomb which is a Lee sphere of radius .

Let be the number of configurations inequivalent under the dihedral group of symmetries of the hexagon.

r3 3( 1)r r

( )N r2 1n r

r( )r

Honeycomb Arrays• Another counting problem is to count all the honeycomb arrays with the requirement that all differences be distinct between the nonattacking bee-Rooks on a honeycomb board of radius .

Let be the total number of honeycomb arrays of radius Let be the number of honeycomb arrays of radius inequivalent under the dihedral group of symmetries of the hexagon.

r

( )H r

2 1r

.r( )h r r

Honeycomb Arrays• Another counting problem is to count all the honeycomb arrays with the requirement that all differences be distinct between the nonattacking bee-Rooks on a honeycomb board of radius .

Let be the total number of honeycomb arrays of radius Let be the number of honeycomb arrays of radius inequivalent under the dihedral group of symmetries of the hexagon.

r

( )H r

2 1r

.r( )h r r

Honeycomb Arrays

Special properties• Nonattacking KingsIn this case we make the Costas array be a configuration of nonattacking chess kings.

In Costas arrays derived from the Welch construction ( ) there will be at least one pair of attacking kings.

One systematic construction of such arrays uses corollary 2.2 – removes , and uses the fact that in Lempel constructions there are no difference vectors parallel to the main diagonal.

7p

(1,2), (2,1)

Special properties• Nonattacking KingsIn this case we make the Costas array be a configuration of nonattacking chess kings.

In Costas arrays derived from the Welch construction ( ) there will be at least one pair of attacking kings.

One systematic construction of such arrays uses corollary 2.2 – removes , and uses the fact that in Lempel constructions there are no difference vectors parallel to the main diagonal.

7p

(1,2), (2,1)

Unsolved problemsDefinitions:

Let be the total number of Costas arrays.

Let be the number of Costas arrays of inequivalent under the dihedral group of symmetries of the square.

Problems:1. for all .

2. is monotonic increasing.

3. . That is has an infinite subsequence which is unbounded above.

4. is monotonic decreasing.

5. as .

6. goes monotonically to 0 as .

( )C n

( )c n

n nn n

( ) 1C n 1n( )C n

( )C nlimsup ( )C n

( ) / !C n n

( ) / !C n n n ( ) / !C n n n

Unsolved problemsProblems:7. .

8. Do any other singly periodic Costas array exists besides the ones given by the Welch construction?

(for and even.)

9. Do honeycombs arrays exist for infinitely many ?

10. Do any Costas arrays exist (for ) which are configurations of nonattacking queens?

16n

( ) / ( )n

C n c n

nn n 1n

The end

Documents

Contents Introduction Related problems Constructions –Welch construction –Lempel construction –Golomb construction Special properties –Periodicity –Nonattacking