ECE4253 Digital Communications Department of Electrical and Computer Engineering - University of New Brunswick, Fredericton, NB, Canada

# Table of Factors for Polynomials in GF(2)

Binary values expressed as polynomials can readily be manipulated using the rules of GF(2). The table below completely factors all polynomials in GF(2) up to degree 6. Irreducible (prime) polynomials are identified (these are unrelated to prime integers.).

 INDEX BINARY POLYNOMIAL FACTORS 2 10 x irreducible 3 11 x+1 irreducible [ LRS ] 4 100 x2 (x)(x) 5 101 x2+1 (x+1)(x+1) 6 110 x2+x (x)(x+1) 7 111 x2+x+1 irreducible [ LRS ] 8 1000 x3 (x)(x)(x) 9 1001 x3+1 (x+1)(x2+x+1) 10 1010 x3+x (x)(x+1)(x+1) 11 1011 x3+x+1 irreducible [ LRS ] 12 1100 x3+x2 (x)(x)(x+1) 13 1101 x3+x2+1 irreducible [ LRS ] 14 1110 x3+x2+x (x)(x2+x+1) 15 1111 x3+x2+x+1 (x+1)(x+1)(x+1) 16 10000 x4 (x)(x)(x)(x) 17 10001 x4+1 (x+1)(x+1)(x+1)(x+1) 18 10010 x4+x (x)(x+1)(x2+x+1) 19 10011 x4+x+1 irreducible [ LRS ] 20 10100 x4+x2 (x)(x)(x+1)(x+1) 21 10101 x4+x2+1 (x2+x+1)(x2+x+1) 22 10110 x4+x2+x (x)(x3+x+1) 23 10111 x4+x2+x+1 (x+1)(x3+x2+1) 24 11000 x4+x3 (x)(x)(x)(x+1) 25 11001 x4+x3+1 irreducible [ LRS ] 26 11010 x4+x3+x (x)(x3+x2+1) 27 11011 x4+x3+x+1 (x+1)(x+1)(x2+x+1) 28 11100 x4+x3+x2 (x)(x)(x2+x+1) 29 11101 x4+x3+x2+1 (x+1)(x3+x+1) 30 11110 x4+x3+x2+x (x)(x+1)(x+1)(x+1) 31 11111 x4+x3+x2+x+1 irreducible [ LRS ] 32 100000 x5 (x)(x)(x)(x)(x) 33 100001 x5+1 (x+1)(x4+x3+x2+x+1) 34 100010 x5+x (x)(x+1)(x+1)(x+1)(x+1) 35 100011 x5+x+1 (x2+x+1)(x3+x2+1) 36 100100 x5+x2 (x)(x)(x+1)(x2+x+1) 37 100101 x5+x2+1 irreducible [ LRS ] 38 100110 x5+x2+x (x)(x4+x+1) 39 100111 x5+x2+x+1 (x+1)(x+1)(x3+x+1) 40 101000 x5+x3 (x)(x)(x)(x+1)(x+1) 41 101001 x5+x3+1 irreducible [ LRS ] 42 101010 x5+x3+x (x)(x2+x+1)(x2+x+1) 43 101011 x5+x3+x+1 (x+1)(x4+x3+1) 44 101100 x5+x3+x2 (x)(x)(x3+x+1) 45 101101 x5+x3+x2+1 (x+1)(x+1)(x+1)(x2+x+1) 46 101110 x5+x3+x2+x (x)(x+1)(x3+x2+1) 47 101111 x5+x3+x2+x+1 irreducible [ LRS ] 48 110000 x5+x4 (x)(x)(x)(x)(x+1) 49 110001 x5+x4+1 (x2+x+1)(x3+x+1) 50 110010 x5+x4+x (x)(x4+x3+1) 51 110011 x5+x4+x+1 (x+1)(x+1)(x+1)(x+1)(x+1) 52 110100 x5+x4+x2 (x)(x)(x3+x2+1) 53 110101 x5+x4+x2+1 (x+1)(x4+x+1) 54 110110 x5+x4+x2+x (x)(x+1)(x+1)(x2+x+1) 55 110111 x5+x4+x2+x+1 irreducible [ LRS ] 56 111000 x5+x4+x3 (x)(x)(x)(x2+x+1) 57 111001 x5+x4+x3+1 (x+1)(x+1)(x3+x2+1) 58 111010 x5+x4+x3+x (x)(x+1)(x3+x+1) 59 111011 x5+x4+x3+x+1 irreducible [ LRS ] 60 111100 x5+x4+x3+x2 (x)(x)(x+1)(x+1)(x+1) 61 111101 x5+x4+x3+x2+1 irreducible [ LRS ] 62 111110 x5+x4+x3+x2+x (x)(x4+x3+x2+x+1) 63 111111 x5+x4+x3+x2+x+1 (x+1)(x2+x+1)(x2+x+1) 64 1000000 x6 (x)(x)(x)(x)(x)(x) 65 1000001 x6+1 (x+1)(x+1)(x2+x+1)(x2+x+1) 66 1000010 x6+x (x)(x+1)(x4+x3+x2+x+1) 67 1000011 x6+x+1 irreducible [ LRS ] 68 1000100 x6+x2 (x)(x)(x+1)(x+1)(x+1)(x+1) 69 1000101 x6+x2+1 (x3+x+1)(x3+x+1) 70 1000110 x6+x2+x (x)(x2+x+1)(x3+x2+1) 71 1000111 x6+x2+x+1 (x+1)(x5+x4+x3+x2+1) 72 1001000 x6+x3 (x)(x)(x)(x+1)(x2+x+1) 73 1001001 x6+x3+1 irreducible [ LRS ] 74 1001010 x6+x3+x (x)(x5+x2+1) 75 1001011 x6+x3+x+1 (x+1)(x+1)(x+1)(x3+x2+1) 76 1001100 x6+x3+x2 (x)(x)(x4+x+1) 77 1001101 x6+x3+x2+1 (x+1)(x5+x4+x3+x+1) 78 1001110 x6+x3+x2+x (x)(x+1)(x+1)(x3+x+1) 79 1001111 x6+x3+x2+x+1 (x2+x+1)(x4+x3+1) 80 1010000 x6+x4 (x)(x)(x)(x)(x+1)(x+1) 81 1010001 x6+x4+1 (x3+x2+1)(x3+x2+1) 82 1010010 x6+x4+x (x)(x5+x3+1) 83 1010011 x6+x4+x+1 (x+1)(x2+x+1)(x3+x+1) 84 1010100 x6+x4+x2 (x)(x)(x2+x+1)(x2+x+1) 85 1010101 x6+x4+x2+1 (x+1)(x+1)(x+1)(x+1)(x+1)(x+1) 86 1010110 x6+x4+x2+x (x)(x+1)(x4+x3+1) 87 1010111 x6+x4+x2+x+1 irreducible [ LRS ] 88 1011000 x6+x4+x3 (x)(x)(x)(x3+x+1) 89 1011001 x6+x4+x3+1 (x+1)(x5+x4+x2+x+1) 90 1011010 x6+x4+x3+x (x)(x+1)(x+1)(x+1)(x2+x+1) 91 1011011 x6+x4+x3+x+1 irreducible [ LRS ] 92 1011100 x6+x4+x3+x2 (x)(x)(x+1)(x3+x2+1) 93 1011101 x6+x4+x3+x2+1 (x2+x+1)(x4+x3+x2+x+1) 94 1011110 x6+x4+x3+x2+x (x)(x5+x3+x2+x+1) 95 1011111 x6+x4+x3+x2+x+1 (x+1)(x+1)(x4+x+1) 96 1100000 x6+x5 (x)(x)(x)(x)(x)(x+1) 97 1100001 x6+x5+1 irreducible [ LRS ] 98 1100010 x6+x5+x (x)(x2+x+1)(x3+x+1) 99 1100011 x6+x5+x+1 (x+1)(x+1)(x4+x3+x2+x+1) 100 1100100 x6+x5+x2 (x)(x)(x4+x3+1) 101 1100101 x6+x5+x2+1 (x+1)(x2+x+1)(x3+x2+1) 102 1100110 x6+x5+x2+x (x)(x+1)(x+1)(x+1)(x+1)(x+1) 103 1100111 x6+x5+x2+x+1 irreducible [ LRS ] 104 1101000 x6+x5+x3 (x)(x)(x)(x3+x2+1) 105 1101001 x6+x5+x3+1 (x+1)(x+1)(x+1)(x3+x+1) 106 1101010 x6+x5+x3+x (x)(x+1)(x4+x+1) 107 1101011 x6+x5+x3+x+1 (x2+x+1)(x2+x+1)(x2+x+1) 108 1101100 x6+x5+x3+x2 (x)(x)(x+1)(x+1)(x2+x+1) 109 1101101 x6+x5+x3+x2+1 irreducible [ LRS ] 110 1101110 x6+x5+x3+x2+x (x)(x5+x4+x2+x+1) 111 1101111 x6+x5+x3+x2+x+1 (x+1)(x5+x2+1) 112 1110000 x6+x5+x4 (x)(x)(x)(x)(x2+x+1) 113 1110001 x6+x5+x4+1 (x+1)(x5+x3+x2+x+1) 114 1110010 x6+x5+x4+x (x)(x+1)(x+1)(x3+x2+1) 115 1110011 x6+x5+x4+x+1 irreducible [ LRS ] 116 1110100 x6+x5+x4+x2 (x)(x)(x+1)(x3+x+1) 117 1110101 x6+x5+x4+x2+1 irreducible [ LRS ] 118 1110110 x6+x5+x4+x2+x (x)(x5+x4+x3+x+1) 119 1110111 x6+x5+x4+x2+x+1 (x+1)(x+1)(x+1)(x+1)(x2+x+1) 120 1111000 x6+x5+x4+x3 (x)(x)(x)(x+1)(x+1)(x+1) 121 1111001 x6+x5+x4+x3+1 (x2+x+1)(x4+x+1) 122 1111010 x6+x5+x4+x3+x (x)(x5+x4+x3+x2+1) 123 1111011 x6+x5+x4+x3+x+1 (x+1)(x5+x3+1) 124 1111100 x6+x5+x4+x3+x2 (x)(x)(x4+x3+x2+x+1) 125 1111101 x6+x5+x4+x3+x2+1 (x+1)(x+1)(x4+x3+1) 126 1111110 x6+x5+x4+x3+x2+x (x)(x+1)(x2+x+1)(x2+x+1) 127 1111111 x6+x5+x4+x3+x2+x+1 (x3+x+1)(x3+x2+1)

 Fri May 17 21:30:29 ADT 2024 Last Updated: 28 SEP 1998 Richard Tervo [ tervo@unb.ca ]