gf2_8.go (23782B)
1 // go-qrcode 2 // Copyright 2014 Tom Harwood 3 4 package reedsolomon 5 6 // Addition, subtraction, multiplication, and division in GF(2^8). 7 // Operations are performed modulo x^8 + x^4 + x^3 + x^2 + 1. 8 9 // http://en.wikipedia.org/wiki/Finite_field_arithmetic 10 11 import "log" 12 13 const ( 14 gfZero = gfElement(0) 15 gfOne = gfElement(1) 16 ) 17 18 var ( 19 gfExpTable = [256]gfElement{ 20 /* 0 - 9 */ 1, 2, 4, 8, 16, 32, 64, 128, 29, 58, 21 /* 10 - 19 */ 116, 232, 205, 135, 19, 38, 76, 152, 45, 90, 22 /* 20 - 29 */ 180, 117, 234, 201, 143, 3, 6, 12, 24, 48, 23 /* 30 - 39 */ 96, 192, 157, 39, 78, 156, 37, 74, 148, 53, 24 /* 40 - 49 */ 106, 212, 181, 119, 238, 193, 159, 35, 70, 140, 25 /* 50 - 59 */ 5, 10, 20, 40, 80, 160, 93, 186, 105, 210, 26 /* 60 - 69 */ 185, 111, 222, 161, 95, 190, 97, 194, 153, 47, 27 /* 70 - 79 */ 94, 188, 101, 202, 137, 15, 30, 60, 120, 240, 28 /* 80 - 89 */ 253, 231, 211, 187, 107, 214, 177, 127, 254, 225, 29 /* 90 - 99 */ 223, 163, 91, 182, 113, 226, 217, 175, 67, 134, 30 /* 100 - 109 */ 17, 34, 68, 136, 13, 26, 52, 104, 208, 189, 31 /* 110 - 119 */ 103, 206, 129, 31, 62, 124, 248, 237, 199, 147, 32 /* 120 - 129 */ 59, 118, 236, 197, 151, 51, 102, 204, 133, 23, 33 /* 130 - 139 */ 46, 92, 184, 109, 218, 169, 79, 158, 33, 66, 34 /* 140 - 149 */ 132, 21, 42, 84, 168, 77, 154, 41, 82, 164, 35 /* 150 - 159 */ 85, 170, 73, 146, 57, 114, 228, 213, 183, 115, 36 /* 160 - 169 */ 230, 209, 191, 99, 198, 145, 63, 126, 252, 229, 37 /* 170 - 179 */ 215, 179, 123, 246, 241, 255, 227, 219, 171, 75, 38 /* 180 - 189 */ 150, 49, 98, 196, 149, 55, 110, 220, 165, 87, 39 /* 190 - 199 */ 174, 65, 130, 25, 50, 100, 200, 141, 7, 14, 40 /* 200 - 209 */ 28, 56, 112, 224, 221, 167, 83, 166, 81, 162, 41 /* 210 - 219 */ 89, 178, 121, 242, 249, 239, 195, 155, 43, 86, 42 /* 220 - 229 */ 172, 69, 138, 9, 18, 36, 72, 144, 61, 122, 43 /* 230 - 239 */ 244, 245, 247, 243, 251, 235, 203, 139, 11, 22, 44 /* 240 - 249 */ 44, 88, 176, 125, 250, 233, 207, 131, 27, 54, 45 /* 250 - 255 */ 108, 216, 173, 71, 142, 1} 46 47 gfLogTable = [256]int{ 48 /* 0 - 9 */ -1, 0, 1, 25, 2, 50, 26, 198, 3, 223, 49 /* 10 - 19 */ 51, 238, 27, 104, 199, 75, 4, 100, 224, 14, 50 /* 20 - 29 */ 52, 141, 239, 129, 28, 193, 105, 248, 200, 8, 51 /* 30 - 39 */ 76, 113, 5, 138, 101, 47, 225, 36, 15, 33, 52 /* 40 - 49 */ 53, 147, 142, 218, 240, 18, 130, 69, 29, 181, 53 /* 50 - 59 */ 194, 125, 106, 39, 249, 185, 201, 154, 9, 120, 54 /* 60 - 69 */ 77, 228, 114, 166, 6, 191, 139, 98, 102, 221, 55 /* 70 - 79 */ 48, 253, 226, 152, 37, 179, 16, 145, 34, 136, 56 /* 80 - 89 */ 54, 208, 148, 206, 143, 150, 219, 189, 241, 210, 57 /* 90 - 99 */ 19, 92, 131, 56, 70, 64, 30, 66, 182, 163, 58 /* 100 - 109 */ 195, 72, 126, 110, 107, 58, 40, 84, 250, 133, 59 /* 110 - 119 */ 186, 61, 202, 94, 155, 159, 10, 21, 121, 43, 60 /* 120 - 129 */ 78, 212, 229, 172, 115, 243, 167, 87, 7, 112, 61 /* 130 - 139 */ 192, 247, 140, 128, 99, 13, 103, 74, 222, 237, 62 /* 140 - 149 */ 49, 197, 254, 24, 227, 165, 153, 119, 38, 184, 63 /* 150 - 159 */ 180, 124, 17, 68, 146, 217, 35, 32, 137, 46, 64 /* 160 - 169 */ 55, 63, 209, 91, 149, 188, 207, 205, 144, 135, 65 /* 170 - 179 */ 151, 178, 220, 252, 190, 97, 242, 86, 211, 171, 66 /* 180 - 189 */ 20, 42, 93, 158, 132, 60, 57, 83, 71, 109, 67 /* 190 - 199 */ 65, 162, 31, 45, 67, 216, 183, 123, 164, 118, 68 /* 200 - 209 */ 196, 23, 73, 236, 127, 12, 111, 246, 108, 161, 69 /* 210 - 219 */ 59, 82, 41, 157, 85, 170, 251, 96, 134, 177, 70 /* 220 - 229 */ 187, 204, 62, 90, 203, 89, 95, 176, 156, 169, 71 /* 230 - 239 */ 160, 81, 11, 245, 22, 235, 122, 117, 44, 215, 72 /* 240 - 249 */ 79, 174, 213, 233, 230, 231, 173, 232, 116, 214, 73 /* 250 - 255 */ 244, 234, 168, 80, 88, 175} 74 ) 75 76 // gfElement is an element in GF(2^8). 77 type gfElement uint8 78 79 // newGFElement creates and returns a new gfElement. 80 func newGFElement(data byte) gfElement { 81 return gfElement(data) 82 } 83 84 // gfAdd returns a + b. 85 func gfAdd(a, b gfElement) gfElement { 86 return a ^ b 87 } 88 89 // gfSub returns a - b. 90 // 91 // Note addition is equivalent to subtraction in GF(2). 92 func gfSub(a, b gfElement) gfElement { 93 return a ^ b 94 } 95 96 // gfMultiply returns a * b. 97 func gfMultiply(a, b gfElement) gfElement { 98 if a == gfZero || b == gfZero { 99 return gfZero 100 } 101 102 return gfExpTable[(gfLogTable[a]+gfLogTable[b])%255] 103 } 104 105 // gfDivide returns a / b. 106 // 107 // Divide by zero results in a panic. 108 func gfDivide(a, b gfElement) gfElement { 109 if a == gfZero { 110 return gfZero 111 } else if b == gfZero { 112 log.Panicln("Divide by zero") 113 } 114 115 return gfMultiply(a, gfInverse(b)) 116 } 117 118 // gfInverse returns the multiplicative inverse of a, a^-1. 119 // 120 // a * a^-1 = 1 121 func gfInverse(a gfElement) gfElement { 122 if a == gfZero { 123 log.Panicln("No multiplicative inverse of 0") 124 } 125 126 return gfExpTable[255-gfLogTable[a]] 127 } 128 129 // a^i | bits | polynomial | decimal 130 // -------------------------------------------------------------------------- 131 // 0 | 000000000 | 0x^8 0x^7 0x^6 0x^5 0x^4 0x^3 0x^2 0x^1 0x^0 | 0 132 // a^0 | 000000001 | 0x^8 0x^7 0x^6 0x^5 0x^4 0x^3 0x^2 0x^1 1x^0 | 1 133 // a^1 | 000000010 | 0x^8 0x^7 0x^6 0x^5 0x^4 0x^3 0x^2 1x^1 0x^0 | 2 134 // a^2 | 000000100 | 0x^8 0x^7 0x^6 0x^5 0x^4 0x^3 1x^2 0x^1 0x^0 | 4 135 // a^3 | 000001000 | 0x^8 0x^7 0x^6 0x^5 0x^4 1x^3 0x^2 0x^1 0x^0 | 8 136 // a^4 | 000010000 | 0x^8 0x^7 0x^6 0x^5 1x^4 0x^3 0x^2 0x^1 0x^0 | 16 137 // a^5 | 000100000 | 0x^8 0x^7 0x^6 1x^5 0x^4 0x^3 0x^2 0x^1 0x^0 | 32 138 // a^6 | 001000000 | 0x^8 0x^7 1x^6 0x^5 0x^4 0x^3 0x^2 0x^1 0x^0 | 64 139 // a^7 | 010000000 | 0x^8 1x^7 0x^6 0x^5 0x^4 0x^3 0x^2 0x^1 0x^0 | 128 140 // a^8 | 000011101 | 0x^8 0x^7 0x^6 0x^5 1x^4 1x^3 1x^2 0x^1 1x^0 | 29 141 // a^9 | 000111010 | 0x^8 0x^7 0x^6 1x^5 1x^4 1x^3 0x^2 1x^1 0x^0 | 58 142 // a^10 | 001110100 | 0x^8 0x^7 1x^6 1x^5 1x^4 0x^3 1x^2 0x^1 0x^0 | 116 143 // a^11 | 011101000 | 0x^8 1x^7 1x^6 1x^5 0x^4 1x^3 0x^2 0x^1 0x^0 | 232 144 // a^12 | 011001101 | 0x^8 1x^7 1x^6 0x^5 0x^4 1x^3 1x^2 0x^1 1x^0 | 205 145 // a^13 | 010000111 | 0x^8 1x^7 0x^6 0x^5 0x^4 0x^3 1x^2 1x^1 1x^0 | 135 146 // a^14 | 000010011 | 0x^8 0x^7 0x^6 0x^5 1x^4 0x^3 0x^2 1x^1 1x^0 | 19 147 // a^15 | 000100110 | 0x^8 0x^7 0x^6 1x^5 0x^4 0x^3 1x^2 1x^1 0x^0 | 38 148 // a^16 | 001001100 | 0x^8 0x^7 1x^6 0x^5 0x^4 1x^3 1x^2 0x^1 0x^0 | 76 149 // a^17 | 010011000 | 0x^8 1x^7 0x^6 0x^5 1x^4 1x^3 0x^2 0x^1 0x^0 | 152 150 // a^18 | 000101101 | 0x^8 0x^7 0x^6 1x^5 0x^4 1x^3 1x^2 0x^1 1x^0 | 45 151 // a^19 | 001011010 | 0x^8 0x^7 1x^6 0x^5 1x^4 1x^3 0x^2 1x^1 0x^0 | 90 152 // a^20 | 010110100 | 0x^8 1x^7 0x^6 1x^5 1x^4 0x^3 1x^2 0x^1 0x^0 | 180 153 // a^21 | 001110101 | 0x^8 0x^7 1x^6 1x^5 1x^4 0x^3 1x^2 0x^1 1x^0 | 117 154 // a^22 | 011101010 | 0x^8 1x^7 1x^6 1x^5 0x^4 1x^3 0x^2 1x^1 0x^0 | 234 155 // a^23 | 011001001 | 0x^8 1x^7 1x^6 0x^5 0x^4 1x^3 0x^2 0x^1 1x^0 | 201 156 // a^24 | 010001111 | 0x^8 1x^7 0x^6 0x^5 0x^4 1x^3 1x^2 1x^1 1x^0 | 143 157 // a^25 | 000000011 | 0x^8 0x^7 0x^6 0x^5 0x^4 0x^3 0x^2 1x^1 1x^0 | 3 158 // a^26 | 000000110 | 0x^8 0x^7 0x^6 0x^5 0x^4 0x^3 1x^2 1x^1 0x^0 | 6 159 // a^27 | 000001100 | 0x^8 0x^7 0x^6 0x^5 0x^4 1x^3 1x^2 0x^1 0x^0 | 12 160 // a^28 | 000011000 | 0x^8 0x^7 0x^6 0x^5 1x^4 1x^3 0x^2 0x^1 0x^0 | 24 161 // a^29 | 000110000 | 0x^8 0x^7 0x^6 1x^5 1x^4 0x^3 0x^2 0x^1 0x^0 | 48 162 // a^30 | 001100000 | 0x^8 0x^7 1x^6 1x^5 0x^4 0x^3 0x^2 0x^1 0x^0 | 96 163 // a^31 | 011000000 | 0x^8 1x^7 1x^6 0x^5 0x^4 0x^3 0x^2 0x^1 0x^0 | 192 164 // a^32 | 010011101 | 0x^8 1x^7 0x^6 0x^5 1x^4 1x^3 1x^2 0x^1 1x^0 | 157 165 // a^33 | 000100111 | 0x^8 0x^7 0x^6 1x^5 0x^4 0x^3 1x^2 1x^1 1x^0 | 39 166 // a^34 | 001001110 | 0x^8 0x^7 1x^6 0x^5 0x^4 1x^3 1x^2 1x^1 0x^0 | 78 167 // a^35 | 010011100 | 0x^8 1x^7 0x^6 0x^5 1x^4 1x^3 1x^2 0x^1 0x^0 | 156 168 // a^36 | 000100101 | 0x^8 0x^7 0x^6 1x^5 0x^4 0x^3 1x^2 0x^1 1x^0 | 37 169 // a^37 | 001001010 | 0x^8 0x^7 1x^6 0x^5 0x^4 1x^3 0x^2 1x^1 0x^0 | 74 170 // a^38 | 010010100 | 0x^8 1x^7 0x^6 0x^5 1x^4 0x^3 1x^2 0x^1 0x^0 | 148 171 // a^39 | 000110101 | 0x^8 0x^7 0x^6 1x^5 1x^4 0x^3 1x^2 0x^1 1x^0 | 53 172 // a^40 | 001101010 | 0x^8 0x^7 1x^6 1x^5 0x^4 1x^3 0x^2 1x^1 0x^0 | 106 173 // a^41 | 011010100 | 0x^8 1x^7 1x^6 0x^5 1x^4 0x^3 1x^2 0x^1 0x^0 | 212 174 // a^42 | 010110101 | 0x^8 1x^7 0x^6 1x^5 1x^4 0x^3 1x^2 0x^1 1x^0 | 181 175 // a^43 | 001110111 | 0x^8 0x^7 1x^6 1x^5 1x^4 0x^3 1x^2 1x^1 1x^0 | 119 176 // a^44 | 011101110 | 0x^8 1x^7 1x^6 1x^5 0x^4 1x^3 1x^2 1x^1 0x^0 | 238 177 // a^45 | 011000001 | 0x^8 1x^7 1x^6 0x^5 0x^4 0x^3 0x^2 0x^1 1x^0 | 193 178 // a^46 | 010011111 | 0x^8 1x^7 0x^6 0x^5 1x^4 1x^3 1x^2 1x^1 1x^0 | 159 179 // a^47 | 000100011 | 0x^8 0x^7 0x^6 1x^5 0x^4 0x^3 0x^2 1x^1 1x^0 | 35 180 // a^48 | 001000110 | 0x^8 0x^7 1x^6 0x^5 0x^4 0x^3 1x^2 1x^1 0x^0 | 70 181 // a^49 | 010001100 | 0x^8 1x^7 0x^6 0x^5 0x^4 1x^3 1x^2 0x^1 0x^0 | 140 182 // a^50 | 000000101 | 0x^8 0x^7 0x^6 0x^5 0x^4 0x^3 1x^2 0x^1 1x^0 | 5 183 // a^51 | 000001010 | 0x^8 0x^7 0x^6 0x^5 0x^4 1x^3 0x^2 1x^1 0x^0 | 10 184 // a^52 | 000010100 | 0x^8 0x^7 0x^6 0x^5 1x^4 0x^3 1x^2 0x^1 0x^0 | 20 185 // a^53 | 000101000 | 0x^8 0x^7 0x^6 1x^5 0x^4 1x^3 0x^2 0x^1 0x^0 | 40 186 // a^54 | 001010000 | 0x^8 0x^7 1x^6 0x^5 1x^4 0x^3 0x^2 0x^1 0x^0 | 80 187 // a^55 | 010100000 | 0x^8 1x^7 0x^6 1x^5 0x^4 0x^3 0x^2 0x^1 0x^0 | 160 188 // a^56 | 001011101 | 0x^8 0x^7 1x^6 0x^5 1x^4 1x^3 1x^2 0x^1 1x^0 | 93 189 // a^57 | 010111010 | 0x^8 1x^7 0x^6 1x^5 1x^4 1x^3 0x^2 1x^1 0x^0 | 186 190 // a^58 | 001101001 | 0x^8 0x^7 1x^6 1x^5 0x^4 1x^3 0x^2 0x^1 1x^0 | 105 191 // a^59 | 011010010 | 0x^8 1x^7 1x^6 0x^5 1x^4 0x^3 0x^2 1x^1 0x^0 | 210 192 // a^60 | 010111001 | 0x^8 1x^7 0x^6 1x^5 1x^4 1x^3 0x^2 0x^1 1x^0 | 185 193 // a^61 | 001101111 | 0x^8 0x^7 1x^6 1x^5 0x^4 1x^3 1x^2 1x^1 1x^0 | 111 194 // a^62 | 011011110 | 0x^8 1x^7 1x^6 0x^5 1x^4 1x^3 1x^2 1x^1 0x^0 | 222 195 // a^63 | 010100001 | 0x^8 1x^7 0x^6 1x^5 0x^4 0x^3 0x^2 0x^1 1x^0 | 161 196 // a^64 | 001011111 | 0x^8 0x^7 1x^6 0x^5 1x^4 1x^3 1x^2 1x^1 1x^0 | 95 197 // a^65 | 010111110 | 0x^8 1x^7 0x^6 1x^5 1x^4 1x^3 1x^2 1x^1 0x^0 | 190 198 // a^66 | 001100001 | 0x^8 0x^7 1x^6 1x^5 0x^4 0x^3 0x^2 0x^1 1x^0 | 97 199 // a^67 | 011000010 | 0x^8 1x^7 1x^6 0x^5 0x^4 0x^3 0x^2 1x^1 0x^0 | 194 200 // a^68 | 010011001 | 0x^8 1x^7 0x^6 0x^5 1x^4 1x^3 0x^2 0x^1 1x^0 | 153 201 // a^69 | 000101111 | 0x^8 0x^7 0x^6 1x^5 0x^4 1x^3 1x^2 1x^1 1x^0 | 47 202 // a^70 | 001011110 | 0x^8 0x^7 1x^6 0x^5 1x^4 1x^3 1x^2 1x^1 0x^0 | 94 203 // a^71 | 010111100 | 0x^8 1x^7 0x^6 1x^5 1x^4 1x^3 1x^2 0x^1 0x^0 | 188 204 // a^72 | 001100101 | 0x^8 0x^7 1x^6 1x^5 0x^4 0x^3 1x^2 0x^1 1x^0 | 101 205 // a^73 | 011001010 | 0x^8 1x^7 1x^6 0x^5 0x^4 1x^3 0x^2 1x^1 0x^0 | 202 206 // a^74 | 010001001 | 0x^8 1x^7 0x^6 0x^5 0x^4 1x^3 0x^2 0x^1 1x^0 | 137 207 // a^75 | 000001111 | 0x^8 0x^7 0x^6 0x^5 0x^4 1x^3 1x^2 1x^1 1x^0 | 15 208 // a^76 | 000011110 | 0x^8 0x^7 0x^6 0x^5 1x^4 1x^3 1x^2 1x^1 0x^0 | 30 209 // a^77 | 000111100 | 0x^8 0x^7 0x^6 1x^5 1x^4 1x^3 1x^2 0x^1 0x^0 | 60 210 // a^78 | 001111000 | 0x^8 0x^7 1x^6 1x^5 1x^4 1x^3 0x^2 0x^1 0x^0 | 120 211 // a^79 | 011110000 | 0x^8 1x^7 1x^6 1x^5 1x^4 0x^3 0x^2 0x^1 0x^0 | 240 212 // a^80 | 011111101 | 0x^8 1x^7 1x^6 1x^5 1x^4 1x^3 1x^2 0x^1 1x^0 | 253 213 // a^81 | 011100111 | 0x^8 1x^7 1x^6 1x^5 0x^4 0x^3 1x^2 1x^1 1x^0 | 231 214 // a^82 | 011010011 | 0x^8 1x^7 1x^6 0x^5 1x^4 0x^3 0x^2 1x^1 1x^0 | 211 215 // a^83 | 010111011 | 0x^8 1x^7 0x^6 1x^5 1x^4 1x^3 0x^2 1x^1 1x^0 | 187 216 // a^84 | 001101011 | 0x^8 0x^7 1x^6 1x^5 0x^4 1x^3 0x^2 1x^1 1x^0 | 107 217 // a^85 | 011010110 | 0x^8 1x^7 1x^6 0x^5 1x^4 0x^3 1x^2 1x^1 0x^0 | 214 218 // a^86 | 010110001 | 0x^8 1x^7 0x^6 1x^5 1x^4 0x^3 0x^2 0x^1 1x^0 | 177 219 // a^87 | 001111111 | 0x^8 0x^7 1x^6 1x^5 1x^4 1x^3 1x^2 1x^1 1x^0 | 127 220 // a^88 | 011111110 | 0x^8 1x^7 1x^6 1x^5 1x^4 1x^3 1x^2 1x^1 0x^0 | 254 221 // a^89 | 011100001 | 0x^8 1x^7 1x^6 1x^5 0x^4 0x^3 0x^2 0x^1 1x^0 | 225 222 // a^90 | 011011111 | 0x^8 1x^7 1x^6 0x^5 1x^4 1x^3 1x^2 1x^1 1x^0 | 223 223 // a^91 | 010100011 | 0x^8 1x^7 0x^6 1x^5 0x^4 0x^3 0x^2 1x^1 1x^0 | 163 224 // a^92 | 001011011 | 0x^8 0x^7 1x^6 0x^5 1x^4 1x^3 0x^2 1x^1 1x^0 | 91 225 // a^93 | 010110110 | 0x^8 1x^7 0x^6 1x^5 1x^4 0x^3 1x^2 1x^1 0x^0 | 182 226 // a^94 | 001110001 | 0x^8 0x^7 1x^6 1x^5 1x^4 0x^3 0x^2 0x^1 1x^0 | 113 227 // a^95 | 011100010 | 0x^8 1x^7 1x^6 1x^5 0x^4 0x^3 0x^2 1x^1 0x^0 | 226 228 // a^96 | 011011001 | 0x^8 1x^7 1x^6 0x^5 1x^4 1x^3 0x^2 0x^1 1x^0 | 217 229 // a^97 | 010101111 | 0x^8 1x^7 0x^6 1x^5 0x^4 1x^3 1x^2 1x^1 1x^0 | 175 230 // a^98 | 001000011 | 0x^8 0x^7 1x^6 0x^5 0x^4 0x^3 0x^2 1x^1 1x^0 | 67 231 // a^99 | 010000110 | 0x^8 1x^7 0x^6 0x^5 0x^4 0x^3 1x^2 1x^1 0x^0 | 134 232 // a^100 | 000010001 | 0x^8 0x^7 0x^6 0x^5 1x^4 0x^3 0x^2 0x^1 1x^0 | 17 233 // a^101 | 000100010 | 0x^8 0x^7 0x^6 1x^5 0x^4 0x^3 0x^2 1x^1 0x^0 | 34 234 // a^102 | 001000100 | 0x^8 0x^7 1x^6 0x^5 0x^4 0x^3 1x^2 0x^1 0x^0 | 68 235 // a^103 | 010001000 | 0x^8 1x^7 0x^6 0x^5 0x^4 1x^3 0x^2 0x^1 0x^0 | 136 236 // a^104 | 000001101 | 0x^8 0x^7 0x^6 0x^5 0x^4 1x^3 1x^2 0x^1 1x^0 | 13 237 // a^105 | 000011010 | 0x^8 0x^7 0x^6 0x^5 1x^4 1x^3 0x^2 1x^1 0x^0 | 26 238 // a^106 | 000110100 | 0x^8 0x^7 0x^6 1x^5 1x^4 0x^3 1x^2 0x^1 0x^0 | 52 239 // a^107 | 001101000 | 0x^8 0x^7 1x^6 1x^5 0x^4 1x^3 0x^2 0x^1 0x^0 | 104 240 // a^108 | 011010000 | 0x^8 1x^7 1x^6 0x^5 1x^4 0x^3 0x^2 0x^1 0x^0 | 208 241 // a^109 | 010111101 | 0x^8 1x^7 0x^6 1x^5 1x^4 1x^3 1x^2 0x^1 1x^0 | 189 242 // a^110 | 001100111 | 0x^8 0x^7 1x^6 1x^5 0x^4 0x^3 1x^2 1x^1 1x^0 | 103 243 // a^111 | 011001110 | 0x^8 1x^7 1x^6 0x^5 0x^4 1x^3 1x^2 1x^1 0x^0 | 206 244 // a^112 | 010000001 | 0x^8 1x^7 0x^6 0x^5 0x^4 0x^3 0x^2 0x^1 1x^0 | 129 245 // a^113 | 000011111 | 0x^8 0x^7 0x^6 0x^5 1x^4 1x^3 1x^2 1x^1 1x^0 | 31 246 // a^114 | 000111110 | 0x^8 0x^7 0x^6 1x^5 1x^4 1x^3 1x^2 1x^1 0x^0 | 62 247 // a^115 | 001111100 | 0x^8 0x^7 1x^6 1x^5 1x^4 1x^3 1x^2 0x^1 0x^0 | 124 248 // a^116 | 011111000 | 0x^8 1x^7 1x^6 1x^5 1x^4 1x^3 0x^2 0x^1 0x^0 | 248 249 // a^117 | 011101101 | 0x^8 1x^7 1x^6 1x^5 0x^4 1x^3 1x^2 0x^1 1x^0 | 237 250 // a^118 | 011000111 | 0x^8 1x^7 1x^6 0x^5 0x^4 0x^3 1x^2 1x^1 1x^0 | 199 251 // a^119 | 010010011 | 0x^8 1x^7 0x^6 0x^5 1x^4 0x^3 0x^2 1x^1 1x^0 | 147 252 // a^120 | 000111011 | 0x^8 0x^7 0x^6 1x^5 1x^4 1x^3 0x^2 1x^1 1x^0 | 59 253 // a^121 | 001110110 | 0x^8 0x^7 1x^6 1x^5 1x^4 0x^3 1x^2 1x^1 0x^0 | 118 254 // a^122 | 011101100 | 0x^8 1x^7 1x^6 1x^5 0x^4 1x^3 1x^2 0x^1 0x^0 | 236 255 // a^123 | 011000101 | 0x^8 1x^7 1x^6 0x^5 0x^4 0x^3 1x^2 0x^1 1x^0 | 197 256 // a^124 | 010010111 | 0x^8 1x^7 0x^6 0x^5 1x^4 0x^3 1x^2 1x^1 1x^0 | 151 257 // a^125 | 000110011 | 0x^8 0x^7 0x^6 1x^5 1x^4 0x^3 0x^2 1x^1 1x^0 | 51 258 // a^126 | 001100110 | 0x^8 0x^7 1x^6 1x^5 0x^4 0x^3 1x^2 1x^1 0x^0 | 102 259 // a^127 | 011001100 | 0x^8 1x^7 1x^6 0x^5 0x^4 1x^3 1x^2 0x^1 0x^0 | 204 260 // a^128 | 010000101 | 0x^8 1x^7 0x^6 0x^5 0x^4 0x^3 1x^2 0x^1 1x^0 | 133 261 // a^129 | 000010111 | 0x^8 0x^7 0x^6 0x^5 1x^4 0x^3 1x^2 1x^1 1x^0 | 23 262 // a^130 | 000101110 | 0x^8 0x^7 0x^6 1x^5 0x^4 1x^3 1x^2 1x^1 0x^0 | 46 263 // a^131 | 001011100 | 0x^8 0x^7 1x^6 0x^5 1x^4 1x^3 1x^2 0x^1 0x^0 | 92 264 // a^132 | 010111000 | 0x^8 1x^7 0x^6 1x^5 1x^4 1x^3 0x^2 0x^1 0x^0 | 184 265 // a^133 | 001101101 | 0x^8 0x^7 1x^6 1x^5 0x^4 1x^3 1x^2 0x^1 1x^0 | 109 266 // a^134 | 011011010 | 0x^8 1x^7 1x^6 0x^5 1x^4 1x^3 0x^2 1x^1 0x^0 | 218 267 // a^135 | 010101001 | 0x^8 1x^7 0x^6 1x^5 0x^4 1x^3 0x^2 0x^1 1x^0 | 169 268 // a^136 | 001001111 | 0x^8 0x^7 1x^6 0x^5 0x^4 1x^3 1x^2 1x^1 1x^0 | 79 269 // a^137 | 010011110 | 0x^8 1x^7 0x^6 0x^5 1x^4 1x^3 1x^2 1x^1 0x^0 | 158 270 // a^138 | 000100001 | 0x^8 0x^7 0x^6 1x^5 0x^4 0x^3 0x^2 0x^1 1x^0 | 33 271 // a^139 | 001000010 | 0x^8 0x^7 1x^6 0x^5 0x^4 0x^3 0x^2 1x^1 0x^0 | 66 272 // a^140 | 010000100 | 0x^8 1x^7 0x^6 0x^5 0x^4 0x^3 1x^2 0x^1 0x^0 | 132 273 // a^141 | 000010101 | 0x^8 0x^7 0x^6 0x^5 1x^4 0x^3 1x^2 0x^1 1x^0 | 21 274 // a^142 | 000101010 | 0x^8 0x^7 0x^6 1x^5 0x^4 1x^3 0x^2 1x^1 0x^0 | 42 275 // a^143 | 001010100 | 0x^8 0x^7 1x^6 0x^5 1x^4 0x^3 1x^2 0x^1 0x^0 | 84 276 // a^144 | 010101000 | 0x^8 1x^7 0x^6 1x^5 0x^4 1x^3 0x^2 0x^1 0x^0 | 168 277 // a^145 | 001001101 | 0x^8 0x^7 1x^6 0x^5 0x^4 1x^3 1x^2 0x^1 1x^0 | 77 278 // a^146 | 010011010 | 0x^8 1x^7 0x^6 0x^5 1x^4 1x^3 0x^2 1x^1 0x^0 | 154 279 // a^147 | 000101001 | 0x^8 0x^7 0x^6 1x^5 0x^4 1x^3 0x^2 0x^1 1x^0 | 41 280 // a^148 | 001010010 | 0x^8 0x^7 1x^6 0x^5 1x^4 0x^3 0x^2 1x^1 0x^0 | 82 281 // a^149 | 010100100 | 0x^8 1x^7 0x^6 1x^5 0x^4 0x^3 1x^2 0x^1 0x^0 | 164 282 // a^150 | 001010101 | 0x^8 0x^7 1x^6 0x^5 1x^4 0x^3 1x^2 0x^1 1x^0 | 85 283 // a^151 | 010101010 | 0x^8 1x^7 0x^6 1x^5 0x^4 1x^3 0x^2 1x^1 0x^0 | 170 284 // a^152 | 001001001 | 0x^8 0x^7 1x^6 0x^5 0x^4 1x^3 0x^2 0x^1 1x^0 | 73 285 // a^153 | 010010010 | 0x^8 1x^7 0x^6 0x^5 1x^4 0x^3 0x^2 1x^1 0x^0 | 146 286 // a^154 | 000111001 | 0x^8 0x^7 0x^6 1x^5 1x^4 1x^3 0x^2 0x^1 1x^0 | 57 287 // a^155 | 001110010 | 0x^8 0x^7 1x^6 1x^5 1x^4 0x^3 0x^2 1x^1 0x^0 | 114 288 // a^156 | 011100100 | 0x^8 1x^7 1x^6 1x^5 0x^4 0x^3 1x^2 0x^1 0x^0 | 228 289 // a^157 | 011010101 | 0x^8 1x^7 1x^6 0x^5 1x^4 0x^3 1x^2 0x^1 1x^0 | 213 290 // a^158 | 010110111 | 0x^8 1x^7 0x^6 1x^5 1x^4 0x^3 1x^2 1x^1 1x^0 | 183 291 // a^159 | 001110011 | 0x^8 0x^7 1x^6 1x^5 1x^4 0x^3 0x^2 1x^1 1x^0 | 115 292 // a^160 | 011100110 | 0x^8 1x^7 1x^6 1x^5 0x^4 0x^3 1x^2 1x^1 0x^0 | 230 293 // a^161 | 011010001 | 0x^8 1x^7 1x^6 0x^5 1x^4 0x^3 0x^2 0x^1 1x^0 | 209 294 // a^162 | 010111111 | 0x^8 1x^7 0x^6 1x^5 1x^4 1x^3 1x^2 1x^1 1x^0 | 191 295 // a^163 | 001100011 | 0x^8 0x^7 1x^6 1x^5 0x^4 0x^3 0x^2 1x^1 1x^0 | 99 296 // a^164 | 011000110 | 0x^8 1x^7 1x^6 0x^5 0x^4 0x^3 1x^2 1x^1 0x^0 | 198 297 // a^165 | 010010001 | 0x^8 1x^7 0x^6 0x^5 1x^4 0x^3 0x^2 0x^1 1x^0 | 145 298 // a^166 | 000111111 | 0x^8 0x^7 0x^6 1x^5 1x^4 1x^3 1x^2 1x^1 1x^0 | 63 299 // a^167 | 001111110 | 0x^8 0x^7 1x^6 1x^5 1x^4 1x^3 1x^2 1x^1 0x^0 | 126 300 // a^168 | 011111100 | 0x^8 1x^7 1x^6 1x^5 1x^4 1x^3 1x^2 0x^1 0x^0 | 252 301 // a^169 | 011100101 | 0x^8 1x^7 1x^6 1x^5 0x^4 0x^3 1x^2 0x^1 1x^0 | 229 302 // a^170 | 011010111 | 0x^8 1x^7 1x^6 0x^5 1x^4 0x^3 1x^2 1x^1 1x^0 | 215 303 // a^171 | 010110011 | 0x^8 1x^7 0x^6 1x^5 1x^4 0x^3 0x^2 1x^1 1x^0 | 179 304 // a^172 | 001111011 | 0x^8 0x^7 1x^6 1x^5 1x^4 1x^3 0x^2 1x^1 1x^0 | 123 305 // a^173 | 011110110 | 0x^8 1x^7 1x^6 1x^5 1x^4 0x^3 1x^2 1x^1 0x^0 | 246 306 // a^174 | 011110001 | 0x^8 1x^7 1x^6 1x^5 1x^4 0x^3 0x^2 0x^1 1x^0 | 241 307 // a^175 | 011111111 | 0x^8 1x^7 1x^6 1x^5 1x^4 1x^3 1x^2 1x^1 1x^0 | 255 308 // a^176 | 011100011 | 0x^8 1x^7 1x^6 1x^5 0x^4 0x^3 0x^2 1x^1 1x^0 | 227 309 // a^177 | 011011011 | 0x^8 1x^7 1x^6 0x^5 1x^4 1x^3 0x^2 1x^1 1x^0 | 219 310 // a^178 | 010101011 | 0x^8 1x^7 0x^6 1x^5 0x^4 1x^3 0x^2 1x^1 1x^0 | 171 311 // a^179 | 001001011 | 0x^8 0x^7 1x^6 0x^5 0x^4 1x^3 0x^2 1x^1 1x^0 | 75 312 // a^180 | 010010110 | 0x^8 1x^7 0x^6 0x^5 1x^4 0x^3 1x^2 1x^1 0x^0 | 150 313 // a^181 | 000110001 | 0x^8 0x^7 0x^6 1x^5 1x^4 0x^3 0x^2 0x^1 1x^0 | 49 314 // a^182 | 001100010 | 0x^8 0x^7 1x^6 1x^5 0x^4 0x^3 0x^2 1x^1 0x^0 | 98 315 // a^183 | 011000100 | 0x^8 1x^7 1x^6 0x^5 0x^4 0x^3 1x^2 0x^1 0x^0 | 196 316 // a^184 | 010010101 | 0x^8 1x^7 0x^6 0x^5 1x^4 0x^3 1x^2 0x^1 1x^0 | 149 317 // a^185 | 000110111 | 0x^8 0x^7 0x^6 1x^5 1x^4 0x^3 1x^2 1x^1 1x^0 | 55 318 // a^186 | 001101110 | 0x^8 0x^7 1x^6 1x^5 0x^4 1x^3 1x^2 1x^1 0x^0 | 110 319 // a^187 | 011011100 | 0x^8 1x^7 1x^6 0x^5 1x^4 1x^3 1x^2 0x^1 0x^0 | 220 320 // a^188 | 010100101 | 0x^8 1x^7 0x^6 1x^5 0x^4 0x^3 1x^2 0x^1 1x^0 | 165 321 // a^189 | 001010111 | 0x^8 0x^7 1x^6 0x^5 1x^4 0x^3 1x^2 1x^1 1x^0 | 87 322 // a^190 | 010101110 | 0x^8 1x^7 0x^6 1x^5 0x^4 1x^3 1x^2 1x^1 0x^0 | 174 323 // a^191 | 001000001 | 0x^8 0x^7 1x^6 0x^5 0x^4 0x^3 0x^2 0x^1 1x^0 | 65 324 // a^192 | 010000010 | 0x^8 1x^7 0x^6 0x^5 0x^4 0x^3 0x^2 1x^1 0x^0 | 130 325 // a^193 | 000011001 | 0x^8 0x^7 0x^6 0x^5 1x^4 1x^3 0x^2 0x^1 1x^0 | 25 326 // a^194 | 000110010 | 0x^8 0x^7 0x^6 1x^5 1x^4 0x^3 0x^2 1x^1 0x^0 | 50 327 // a^195 | 001100100 | 0x^8 0x^7 1x^6 1x^5 0x^4 0x^3 1x^2 0x^1 0x^0 | 100 328 // a^196 | 011001000 | 0x^8 1x^7 1x^6 0x^5 0x^4 1x^3 0x^2 0x^1 0x^0 | 200 329 // a^197 | 010001101 | 0x^8 1x^7 0x^6 0x^5 0x^4 1x^3 1x^2 0x^1 1x^0 | 141 330 // a^198 | 000000111 | 0x^8 0x^7 0x^6 0x^5 0x^4 0x^3 1x^2 1x^1 1x^0 | 7 331 // a^199 | 000001110 | 0x^8 0x^7 0x^6 0x^5 0x^4 1x^3 1x^2 1x^1 0x^0 | 14 332 // a^200 | 000011100 | 0x^8 0x^7 0x^6 0x^5 1x^4 1x^3 1x^2 0x^1 0x^0 | 28 333 // a^201 | 000111000 | 0x^8 0x^7 0x^6 1x^5 1x^4 1x^3 0x^2 0x^1 0x^0 | 56 334 // a^202 | 001110000 | 0x^8 0x^7 1x^6 1x^5 1x^4 0x^3 0x^2 0x^1 0x^0 | 112 335 // a^203 | 011100000 | 0x^8 1x^7 1x^6 1x^5 0x^4 0x^3 0x^2 0x^1 0x^0 | 224 336 // a^204 | 011011101 | 0x^8 1x^7 1x^6 0x^5 1x^4 1x^3 1x^2 0x^1 1x^0 | 221 337 // a^205 | 010100111 | 0x^8 1x^7 0x^6 1x^5 0x^4 0x^3 1x^2 1x^1 1x^0 | 167 338 // a^206 | 001010011 | 0x^8 0x^7 1x^6 0x^5 1x^4 0x^3 0x^2 1x^1 1x^0 | 83 339 // a^207 | 010100110 | 0x^8 1x^7 0x^6 1x^5 0x^4 0x^3 1x^2 1x^1 0x^0 | 166 340 // a^208 | 001010001 | 0x^8 0x^7 1x^6 0x^5 1x^4 0x^3 0x^2 0x^1 1x^0 | 81 341 // a^209 | 010100010 | 0x^8 1x^7 0x^6 1x^5 0x^4 0x^3 0x^2 1x^1 0x^0 | 162 342 // a^210 | 001011001 | 0x^8 0x^7 1x^6 0x^5 1x^4 1x^3 0x^2 0x^1 1x^0 | 89 343 // a^211 | 010110010 | 0x^8 1x^7 0x^6 1x^5 1x^4 0x^3 0x^2 1x^1 0x^0 | 178 344 // a^212 | 001111001 | 0x^8 0x^7 1x^6 1x^5 1x^4 1x^3 0x^2 0x^1 1x^0 | 121 345 // a^213 | 011110010 | 0x^8 1x^7 1x^6 1x^5 1x^4 0x^3 0x^2 1x^1 0x^0 | 242 346 // a^214 | 011111001 | 0x^8 1x^7 1x^6 1x^5 1x^4 1x^3 0x^2 0x^1 1x^0 | 249 347 // a^215 | 011101111 | 0x^8 1x^7 1x^6 1x^5 0x^4 1x^3 1x^2 1x^1 1x^0 | 239 348 // a^216 | 011000011 | 0x^8 1x^7 1x^6 0x^5 0x^4 0x^3 0x^2 1x^1 1x^0 | 195 349 // a^217 | 010011011 | 0x^8 1x^7 0x^6 0x^5 1x^4 1x^3 0x^2 1x^1 1x^0 | 155 350 // a^218 | 000101011 | 0x^8 0x^7 0x^6 1x^5 0x^4 1x^3 0x^2 1x^1 1x^0 | 43 351 // a^219 | 001010110 | 0x^8 0x^7 1x^6 0x^5 1x^4 0x^3 1x^2 1x^1 0x^0 | 86 352 // a^220 | 010101100 | 0x^8 1x^7 0x^6 1x^5 0x^4 1x^3 1x^2 0x^1 0x^0 | 172 353 // a^221 | 001000101 | 0x^8 0x^7 1x^6 0x^5 0x^4 0x^3 1x^2 0x^1 1x^0 | 69 354 // a^222 | 010001010 | 0x^8 1x^7 0x^6 0x^5 0x^4 1x^3 0x^2 1x^1 0x^0 | 138 355 // a^223 | 000001001 | 0x^8 0x^7 0x^6 0x^5 0x^4 1x^3 0x^2 0x^1 1x^0 | 9 356 // a^224 | 000010010 | 0x^8 0x^7 0x^6 0x^5 1x^4 0x^3 0x^2 1x^1 0x^0 | 18 357 // a^225 | 000100100 | 0x^8 0x^7 0x^6 1x^5 0x^4 0x^3 1x^2 0x^1 0x^0 | 36 358 // a^226 | 001001000 | 0x^8 0x^7 1x^6 0x^5 0x^4 1x^3 0x^2 0x^1 0x^0 | 72 359 // a^227 | 010010000 | 0x^8 1x^7 0x^6 0x^5 1x^4 0x^3 0x^2 0x^1 0x^0 | 144 360 // a^228 | 000111101 | 0x^8 0x^7 0x^6 1x^5 1x^4 1x^3 1x^2 0x^1 1x^0 | 61 361 // a^229 | 001111010 | 0x^8 0x^7 1x^6 1x^5 1x^4 1x^3 0x^2 1x^1 0x^0 | 122 362 // a^230 | 011110100 | 0x^8 1x^7 1x^6 1x^5 1x^4 0x^3 1x^2 0x^1 0x^0 | 244 363 // a^231 | 011110101 | 0x^8 1x^7 1x^6 1x^5 1x^4 0x^3 1x^2 0x^1 1x^0 | 245 364 // a^232 | 011110111 | 0x^8 1x^7 1x^6 1x^5 1x^4 0x^3 1x^2 1x^1 1x^0 | 247 365 // a^233 | 011110011 | 0x^8 1x^7 1x^6 1x^5 1x^4 0x^3 0x^2 1x^1 1x^0 | 243 366 // a^234 | 011111011 | 0x^8 1x^7 1x^6 1x^5 1x^4 1x^3 0x^2 1x^1 1x^0 | 251 367 // a^235 | 011101011 | 0x^8 1x^7 1x^6 1x^5 0x^4 1x^3 0x^2 1x^1 1x^0 | 235 368 // a^236 | 011001011 | 0x^8 1x^7 1x^6 0x^5 0x^4 1x^3 0x^2 1x^1 1x^0 | 203 369 // a^237 | 010001011 | 0x^8 1x^7 0x^6 0x^5 0x^4 1x^3 0x^2 1x^1 1x^0 | 139 370 // a^238 | 000001011 | 0x^8 0x^7 0x^6 0x^5 0x^4 1x^3 0x^2 1x^1 1x^0 | 11 371 // a^239 | 000010110 | 0x^8 0x^7 0x^6 0x^5 1x^4 0x^3 1x^2 1x^1 0x^0 | 22 372 // a^240 | 000101100 | 0x^8 0x^7 0x^6 1x^5 0x^4 1x^3 1x^2 0x^1 0x^0 | 44 373 // a^241 | 001011000 | 0x^8 0x^7 1x^6 0x^5 1x^4 1x^3 0x^2 0x^1 0x^0 | 88 374 // a^242 | 010110000 | 0x^8 1x^7 0x^6 1x^5 1x^4 0x^3 0x^2 0x^1 0x^0 | 176 375 // a^243 | 001111101 | 0x^8 0x^7 1x^6 1x^5 1x^4 1x^3 1x^2 0x^1 1x^0 | 125 376 // a^244 | 011111010 | 0x^8 1x^7 1x^6 1x^5 1x^4 1x^3 0x^2 1x^1 0x^0 | 250 377 // a^245 | 011101001 | 0x^8 1x^7 1x^6 1x^5 0x^4 1x^3 0x^2 0x^1 1x^0 | 233 378 // a^246 | 011001111 | 0x^8 1x^7 1x^6 0x^5 0x^4 1x^3 1x^2 1x^1 1x^0 | 207 379 // a^247 | 010000011 | 0x^8 1x^7 0x^6 0x^5 0x^4 0x^3 0x^2 1x^1 1x^0 | 131 380 // a^248 | 000011011 | 0x^8 0x^7 0x^6 0x^5 1x^4 1x^3 0x^2 1x^1 1x^0 | 27 381 // a^249 | 000110110 | 0x^8 0x^7 0x^6 1x^5 1x^4 0x^3 1x^2 1x^1 0x^0 | 54 382 // a^250 | 001101100 | 0x^8 0x^7 1x^6 1x^5 0x^4 1x^3 1x^2 0x^1 0x^0 | 108 383 // a^251 | 011011000 | 0x^8 1x^7 1x^6 0x^5 1x^4 1x^3 0x^2 0x^1 0x^0 | 216 384 // a^252 | 010101101 | 0x^8 1x^7 0x^6 1x^5 0x^4 1x^3 1x^2 0x^1 1x^0 | 173 385 // a^253 | 001000111 | 0x^8 0x^7 1x^6 0x^5 0x^4 0x^3 1x^2 1x^1 1x^0 | 71 386 // a^254 | 010001110 | 0x^8 1x^7 0x^6 0x^5 0x^4 1x^3 1x^2 1x^1 0x^0 | 142 387 // a^255 | 000000001 | 0x^8 0x^7 0x^6 0x^5 0x^4 0x^3 0x^2 0x^1 1x^0 | 1