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

# Factors of xN+1 in GF(2)

Binary values expressed as polynomials can readily be manipulated using the rules of GF(2). The table below shows the irreducible factors of selected GF(2) polynomials of the form xN+1.

Because addition and subtraction are performed identically in GF(2), this is also a table of factors of xN−1.

Note for example that (1001)2 = (1000001) or (x3+1)2 = (x6+1) and in general (xk+1)2 = (x2k+1), such that factors of a term N=k are found duplicated for N=2k.

 N Irreducible Factors of xN+1 1 (11) 2 (11)2 3 (11)(111) 4 (11)4 5 (11)(11111) 6 (11)2(111)2 7 (11)(1011)(1101) 8 (11)8 9 (11)(111)(1001001) 10 (11)2(11111)2 11 (11)(11111111111) 12 (11)4(111)4 13 (11)(1111111111111) 14 (11)2(1011)2(1101)2 15 (11)(111)(10011)(11001)(11111) 16 (11)16 17 (11)(100111001)(111010111) 18 (11)2(111)2(1001001)2 19 (11)(1111111111111111111) 20 (11)4(11111)4 21 (11)(111)(1011)(1101)(1010111)(1110101) 22 (11)2(11111111111)2 23 (11)(101011100011)(110001110101) 24 (11)8(111)8 25 (11)(11111)(100001000010000100001) 26 (11)2(1111111111111)2 27 (11)(111)(1001001)(1000000001000000001) 28 (11)4(1011)4(1101)4 29 (11)(11111111111111111111111111111) 30 (11)2(111)2(10011)2(11001)2(11111)2 31 (11)(100101)(101001)(101111)(110111)(111011)(111101) 33 (11)(111)(10010101001)(11000100011)(11111111111) 35 (11)(1011)(1101)(11111)(1011110010111)(1110100111101) 37 (11)(1111111111111111111111111111111111111) 39 (11)(111)(1011110001111)(1111000111101)(1111111111111) 41 (11)(101111100111001111101)(110110100111001011011) 43 (11)(100111111111001)(101010010010101)(110100010001011) 45 (11)(111)(10011)(11001)(11111)(1001001)(1000000001001)(1001000000001) 47 (11)(100011000111011011101111)(111101110110111000110001) 49 (11)(1011)(1101)(1000000000000010000001)(1000000100000000000001) 51 (11)(111)(100011011)(100111001)(110011111)(110110001)(111010111)(111110011) 53 (11)(11111111111111111111111111111111111111111111111111111) 55 (11)(11111)(11111111111)(101101101001011100111)(111001110100101101101) 57 (11)(111)(1011100001000011101)(1111011101011101111)(1111111111111111111) 59 (11)(11111111111111111111111111111111111111111111111111111111111) 61 (11)(1111111111111111111111111111111111111111111111111111111111111) 63 (11)(111)(1011)(1101)(1000011)(1001001)(1010111)(1011011)(1100001)(1100111)(1101101)(1110011)(1110101) 65 (11)(11111)(1000111110001)(1010011100101)(1011101011101)(1101011101011)(1111111111111) 67 (11)(1111111111111111111111111111111111111111111111111111111111111111111) 69 (11)(111)(101011100011)(110001110101)(10100110011000001100111)(11100110000011001100101) 71 (11)(101000011111000000100010000110110011)(110011011000010001000000111110000101) 73 (11)(1000000011)(1000010111)(1001001011)(1001100101)(1010011001)(1100000001)(1101001001)(1110100001) 75 (11)(111)(10011)(11001)(11111)(100000000000000100001)(100001000000000000001)(100001000010000100001) 77 (11)(1011)(1101)(11111111111)(1011100101110010111001110010111)(1110100111001110100111010011101) 79 (11)(1001100011101111001111010110100000110111)(1110110000010110101111001111011100011001) 81 (11)(111)(1001001)(1000000001000000001)(1000000000000000000000000001000000000000000000000000001) 83 (11)(11111111111111111111111111111111111111111111111111111111111111111111111111111111111) 85 (11)(11111)(100111001)(100111111)(101110111)(101111011)(110001011)(110100011)(110111101)(111010111)(111011101)(111111001) 87 (11)(111)(10110011110111110001100011111)(11111000110001111101111001101)(11111111111111111111111111111) 89 (11)(100011000011)(100100110111)(100111101111)(110000110001)(110111111111)(111011001001)(111101111001)(111111111011) 91 (11)(1011)(1101)(1000101111001)(1001111010001)(1011000000011)(1100000001101)(1101001111011)(1101111001011)(1111111111111) 93 (11)(111)(100101)(101001)(101111)(110111)(111011)(111101)(10000110101)(10100001011)(10101100001)(11010000101)(11010100111)(11100101011) 95 (11)(11111)(1111111111111111111)(1000000110100011101111101001011100011)(1100011101001011111011100010110000001) 97 (11)(1000010110101000100100001000010010001010110100001)(1111100100001101101100011100011011011000010011111) 99 (11)(111)(1001001)(10010101001)(11000100011)(11111111111)(1000000001000001000001000000001)(1001000000000001000000000001001) 101 (11)(11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111) 103 (11)(1011000111000010100111110100000111101111001100001011)(1101000011001111011110000010111110010100001110001101) 105 (11)(111)(1011)(1101)(10011)(11001)(11111)(1010111)(1110101)(1000101101101)(1001000111011)(1011011010001)(1011110010111)(1101110001001)(1110100111101) 127 (11)(10000011)(10001001)(10001111)(10010001)(10011101)(10100111)(10101011)(10111001)(10111111)(11000001)(11001011)(11010011)(11010101)(11100101)(11101111)(11110001)(11110111)(11111101) 255 (11)(111)(10011)(11001)(11111)(100011011)(100011101)(100101011)(100101101)(100111001)(100111111)(101001101)(101011111)(101100011)(101100101)(101101001)(101110001)(101110111)(101111011)(110000111)(110001011)(110001101)(110011111)(110100011)(110101001)(110110001)(110111101)(111000011)(111001111)(111010111)(111011101)(111100111)(111110011)(111110101)(111111001) 511 (11)(1011)(1101)(1000000011)(1000010001)(1000010111)(1000011011)(1000100001)(1000101101)(1000110011)(1001001011)(1001011001)(1001011111)(1001100101)(1001101001)(1001101111)(1001110111)(1001111101)(1010000111)(1010010101)(1010011001)(1010100011)(1010100101)(1010101111)(1010110111)(1010111101)(1011001111)(1011010001)(1011011011)(1011110101)(1011111001)(1100000001)(1100010011)(1100010101)(1100011111)(1100100011)(1100110001)(1100111011)(1101001001)(1101001111)(1101011011)(1101100001)(1101101011)(1101101101)(1101110011)(1101111111)(1110000101)(1110001111)(1110100001)(1110110101)(1110111001)(1111000111)(1111001011)(1111001101)(1111010101)(1111011001)(1111100011)(1111101001)(1111111011) 1023 (11)(111)(100101)(101001)(101111)(110111)(111011)(111101)(10000001001)(10000001111)(10000011011)(10000011101)(10000100111)(10000101101)(10000110101)(10001000111)(10001010011)(10001100011)(10001100101)(10001101111)(10010000001)(10010001011)(10010011001)(10010101001)(10010101111)(10011000101)(10011001001)(10011010111)(10011100111)(10011101101)(10011110011)(10011111111)(10100001011)(10100001101)(10100011001)(10100011111)(10100100011)(10100110001)(10100111101)(10101000011)(10101010111)(10101100001)(10101100111)(10101101011)(10110000101)(10110001111)(10110010111)(10110011011)(10110100001)(10110101011)(10110111001)(10111000001)(10111000111)(10111100101)(10111110111)(10111111011)(11000010011)(11000010101)(11000100011)(11000100101)(11000110001)(11000110111)(11001000011)(11001001111)(11001010001)(11001011011)(11001111001)(11001111111)(11010000101)(11010001001)(11010100111)(11010101101)(11010110101)(11010111111)(11011000001)(11011001101)(11011010011)(11011011111)(11011110111)(11011111101)(11100001111)(11100010001)(11100010111)(11100011101)(11100100001)(11100101011)(11100110101)(11100111001)(11101000111)(11101001101)(11101010101)(11101011001)(11101100011)(11101111011)(11101111101)(11110000001)(11110000111)(11110001101)(11110010011)(11110101001)(11110110001)(11111000101)(11111011011)(11111101011)(11111110011)(11111111001)(11111111111)

 Fri May 17 22:27:19 ADT 2024 Last Updated: 26 MAR 2023 Richard Tervo [ tervo@unb.ca ]