Editing Linear-feedback shift register (section)

== Example polynomials for maximal LFSRs ==
The following table lists examples of maximal-length feedback polynomials ([[Primitive polynomial (field theory)|primitive polynomials]]) for shift-register lengths up to 24. The formalism for maximum-length LFSRs was developed by [[Solomon W. Golomb]] in his 1967 book.<ref>{{cite book |last1=Golomb |first1=Solomon W. |title=Shift register sequences |date=1967 |publisher=Aegean Park Press |location=Laguna Hills, Calif. |isbn=978-0894120480}}</ref> The number of different [[Primitive polynomial (field theory)|primitive polynomials]] grows exponentially with shift-register length and can be calculated exactly using [[Euler's totient function]]<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Primitive Polynomial|url=https://mathworld.wolfram.com/PrimitivePolynomial.html|access-date=2021-04-27|website=mathworld.wolfram.com|language=en}}</ref> {{OEIS|A011260}}.

{|class="wikitable" style="text-align:right"
|- 
!Bits (n)
!Feedback polynomial
!Taps
!Taps ([[Hexadecimal|hex]])
!Period (<math>2^n - 1</math>)
|- style="font-family:monospace;"
! 2
|<math>x^2 + x + 1</math>
|11
|0x3
| 3
|- style="font-family:monospace;"
! 3
|<math>x^3 + x^2 + 1</math>
|110
|0x6
| 7
|- style="font-family:monospace;"
! 4
|<math>x^4 + x^3 + 1</math>
|1100
|0xC
| 15
|- style="font-family:monospace;"
! 5
|<math>x^{ 5 }+x^{ 3 }+1</math>
|10100
|0x14
| 31
|- style="font-family:monospace;"
! 6
|<math>x^{ 6 }+x^{ 5 }+1</math>
|110000
|0x30
| 63
|- style="font-family:monospace;"
! 7
|<math>x^{ 7 }+x^{ 6 }+1</math>
|1100000
|0x60
| 127
|- style="font-family:monospace;"
! 8
|<math>x^{ 8 }+x^{ 6 }+x^{ 5 }+x^{ 4 }+1</math>
|10111000
|0xB8
| 255
|- style="font-family:monospace;"
! 9
|<math>x^{ 9 }+x^{ 5 }+1</math>
|100010000
|0x110
| 511
|- style="font-family:monospace;"
! 10
|<math>x^{ 10 }+x^{ 7 }+1</math>
|1001000000
|0x240
| 1,023
|- style="font-family:monospace;"
! 11
|<math>x^{ 11 }+x^{ 9 }+1</math>
|10100000000
|0x500
| 2,047
|- style="font-family:monospace;"
! 12
|<math>x^{ 12 }+x^{ 11 }+x^{ 10 }+x^{ 4 }+1</math>
|111000001000
|0xE08
| 4,095
|- style="font-family:monospace;"
! 13
|<math>x^{ 13 }+x^{ 12 }+x^{ 11 }+x^{ 8 }+1</math>
|1110010000000
|0x1C80
| 8,191
|- style="font-family:monospace;"
! 14
|<math>x^{ 14 }+x^{ 13 }+x^{ 12 }+x^{ 2 }+1</math>
|11100000000010
|0x3802
| 16,383
|- style="font-family:monospace;"
! 15
|<math>x^{ 15 }+x^{ 14 }+1</math>
|110000000000000
|0x6000
| 32,767
|- style="font-family:monospace;"
! 16
|<math>x^{ 16 }+x^{ 15 }+x^{ 13 }+x^{ 4 }+1</math>
|1101000000001000
|0xD008
| 65,535
|- style="font-family:monospace;"
! 17
|<math>x^{ 17 }+x^{ 14 }+1</math>
|10010000000000000
|0x12000
| 131,071
|- style="font-family:monospace;"
! 18
|<math>x^{ 18 }+x^{ 11 }+1</math>
|100000010000000000
|0x20400
| 262,143
|- style="font-family:monospace;"
! 19
|<math>x^{ 19 }+x^{ 18 }+x^{ 17 }+x^{ 14 }+1</math>
|1110010000000000000
|0x72000
| 524,287
|- style="font-family:monospace;"
! 20
|<math>x^{ 20 }+x^{ 17 }+1</math>
|10010000000000000000
|0x90000
| 1,048,575
|- style="font-family:monospace;"
! 21
|<math>x^{ 21 }+x^{ 19 }+1</math>
|101000000000000000000
|0x140000
| 2,097,151
|- style="font-family:monospace;"
! 22
|<math>x^{ 22 }+x^{ 21 }+1</math>
|1100000000000000000000
|0x300000
| 4,194,303
|- style="font-family:monospace;"
! 23
|<math>x^{ 23 }+x^{ 18 }+1</math>
|10000100000000000000000
|0x420000
| 8,388,607
|- style="font-family:monospace;"
! 24
|<math>x^{ 24 }+x^{ 23 }+x^{ 22 }+x^{ 17 }+1</math>
|111000010000000000000000
|0xE10000
| 16,777,215
|}