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Table 2 Distribution of cross-correlation values between family A and p p × p equivalent Hadamard variant

From: Extended families of 2D arrays with near optimal auto and low cross-correlation

p = 4k-1

Frequency of increasing cross-correlation level

 

1

2

3

4

5

6

7

8

9

10

7

0.5182

0.4519

0.0299

0

0

0

    

11

0.4324

0.4879

0.0774

0.0023

0

0

0

   

19

0.3443

0.4769

0.1556

0.0222

0.001

0

0

0

  

23

0.3158

0.4639

0.1812

0.0358

0.0032

0.0001

6E-06

0

0

 

31

0.2755

0.4354

0.2135

0.0635

0.011

0.001

5E-05

9E-07

0

 

43

0.2356

0.396

0.2381

0.0979

0.0275

0.005

0.0006

0.0001

0

0

47

0.2283

0.3857

0.2373

0.1064

0.0335

0.0075

0.0013

0.0001

  

59

0.2041

0.3571

0.2416

0.1271

0.0508

0.0154

0.0033

0.0005

0

0

67

0.1913

0.3419

0.2417

0.1361

0.0606

0.0211

0.0058

0.0012

0.0002

4E-05

71

0.1862

0.3342

0.2416

0.1397

0.0648

0.0243

0.0071

0.0017

0.0003

0

79

0.1766

0.3206

0.2393

0.1464

0.0736

0.0301

0.01

0.0027

0.0006

0.0001

83

0.1721

0.3148

0.2386

0.1486

0.0764

0.0337

0.0115

0.0034

0.0079

0.0014

103

0.1546

0.2888

0.2302

0.1567

0.0924

0.0467

0.0201

0.0072

0.0024

0.0006

127

0.1404

0.2637

0.2189

0.1614

0.1042

0.06

0.0302

0.0135

0.0052

0.0018

131

0.1375

0.2606

0.2166

0.1622

0.1059

0.0626

0.0315

0.0144

0.0058

0.002

139

0.1342

0.2531

0.2147

0.1618

0.109

0.0645

0.0349

0.0166

0.0072

0.0026

151

0.1278

0.2438

0.2109

0.1586

0.114

0.0718

0.0395

0.0189

0.0094

0.0035

199

0.1118

0.2168

0.1927

0.1565

0.119

0.0832

0.0537

0.0325

0.0178

0.0091

  1. The correlation is optimally low in the second left column and increases with each column to the right. The entries in italics mark the level at each p where the cross-correlation becomes an autocorrelation. The sum of the entries on each row is normalized to 1. Bold entries mark the array size p where the maximum frequency for level i occurs