This page demonstrates how to generate ECC blocks by performing polynomial divison on a message polynomial. Just enter the coefficients of the message polynomial and the desired number of ECC blocks, then click Perform Division.

If the steps are unclear, please read the error correction generation section of the tutorial, which includes a detailed explanation of these steps.

Enter the coefficients of the message polynomial, separated by commas (no spaces):
(example: 32,91,11,120,209,114,220,77,67,64,236,17,236 )

Enter the desired number of ECC blocks:
(example: 13 )

Perform Division

Polynomial Division Steps

The first step to the division is to prepare the message polynomial for the division. The full message polynomial is:

67x¹⁴ + 85x¹³ + 70x¹² + 134x¹¹ + 87x¹⁰ + 38x⁹ + 85x⁸ + 194x⁷ + 119x⁶ + 50x⁵ + 6x⁴ + 18x³ + 6x² + 103x¹ + 38

To make sure that the exponent of the lead term doesn't become too small during the division, multiply the message polynomial by xⁿ where n is the number of error correction codewords that are needed. In this case n is 18, for 18 error correction codewords, so multiply the message polynomial by x¹⁸, which gives us:

67x³² + 85x³¹ + 70x³⁰ + 134x²⁹ + 87x²⁸ + 38x²⁷ + 85x²⁶ + 194x²⁵ + 119x²⁴ + 50x²³ + 6x²² + 18x²¹ + 6x²⁰ + 103x¹⁹ + 38x¹⁸

The lead term of the generator polynomial should also have the same exponent, so multiply by x¹⁴ to get

α⁰x³² + α²¹⁵x³¹ + α²³⁴x³⁰ + α¹⁵⁸x²⁹ + α⁹⁴x²⁸ + α¹⁸⁴x²⁷ + α⁹⁷x²⁶ + α¹¹⁸x²⁵ + α¹⁷⁰x²⁴ + α⁷⁹x²³ + α¹⁸⁷x²² + α¹⁵²x²¹ + α¹⁴⁸x²⁰ + α²⁵²x¹⁹ + α¹⁷⁹x¹⁸ + α⁵x¹⁷ + α⁹⁸x¹⁶ + α⁹⁶x¹⁵ + α¹⁵³x¹⁴

Now it is possible to perform the repeated division steps. The number of steps in the division must equal the number of terms in the message polynomial. In this case, the division will take 15 steps to complete. This will result in a remainder that has 18 terms. These terms will be the 18 error correction codewords that are required.

Step 1a: Multiply the Generator Polynomial by the Lead Term of the Message Polynomial

The first step is to multiply the generator polynomial by the lead term of the message polynomial. The lead term in this case is 67x³². Since alpha notation makes it easier to perform the multiplication, it is recommended to convert 67x³² to alpha notation. According to the log antilog table, for the integer value 67, the alpha exponent is 98. Therefore 67 = α⁹⁸. Multiply the generator polynomial by α⁹⁸:

The exponents of the alphas are added together. In this case, at least one of the exponents is larger than 255, so perform modulo 255 as follows:

α⁹⁸x³² + α^{(313 % 255)}x³¹ + α^{(332 % 255)}x³⁰ + α^{(256 % 255)}x²⁹ + α¹⁹²x²⁸ + α^{(282 % 255)}x²⁷ + α¹⁹⁵x²⁶ + α²¹⁶x²⁵ + α^{(268 % 255)}x²⁴ + α¹⁷⁷x²³ + α^{(285 % 255)}x²² + α²⁵⁰x²¹ + α²⁴⁶x²⁰ + α^{(350 % 255)}x¹⁹ + α^{(277 % 255)}x¹⁸ + α¹⁰³x¹⁷ + α¹⁹⁶x¹⁶ + α¹⁹⁴x¹⁵ + α²⁵¹x¹⁴

The result is:

α⁹⁸x³² + α⁵⁸x³¹ + α⁷⁷x³⁰ + α¹x²⁹ + α¹⁹²x²⁸ + α²⁷x²⁷ + α¹⁹⁵x²⁶ + α²¹⁶x²⁵ + α¹³x²⁴ + α¹⁷⁷x²³ + α³⁰x²² + α²⁵⁰x²¹ + α²⁴⁶x²⁰ + α⁹⁵x¹⁹ + α²²x¹⁸ + α¹⁰³x¹⁷ + α¹⁹⁶x¹⁶ + α¹⁹⁴x¹⁵ + α²⁵¹x¹⁴

Now, convert this to integer notation:

67x³² + 105x³¹ + 60x³⁰ + 2x²⁹ + 130x²⁸ + 12x²⁷ + 100x²⁶ + 195x²⁵ + 135x²⁴ + 219x²³ + 96x²² + 108x²¹ + 207x²⁰ + 226x¹⁹ + 234x¹⁸ + 136x¹⁷ + 200x¹⁶ + 50x¹⁵ + 216x¹⁴

Step 1b: XOR the result with the message polynomial

Since this is the first division step, XOR the result from 1a with the message polynomial.

(67 ⊕ 67)x³² + (85 ⊕ 105)x³¹ + (70 ⊕ 60)x³⁰ + (134 ⊕ 2)x²⁹ + (87 ⊕ 130)x²⁸ + (38 ⊕ 12)x²⁷ + (85 ⊕ 100)x²⁶ + (194 ⊕ 195)x²⁵ + (119 ⊕ 135)x²⁴ + (50 ⊕ 219)x²³ + (6 ⊕ 96)x²² + (18 ⊕ 108)x²¹ + (6 ⊕ 207)x²⁰ + (103 ⊕ 226)x¹⁹ + (38 ⊕ 234)x¹⁸ + (0 ⊕ 136)x¹⁷ + (0 ⊕ 200)x¹⁶ + (0 ⊕ 50)x¹⁵ + (0 ⊕ 216)x¹⁴

The result is:

0x³² + 60x³¹ + 122x³⁰ + 132x²⁹ + 213x²⁸ + 42x²⁷ + 49x²⁶ + 1x²⁵ + 240x²⁴ + 233x²³ + 102x²² + 126x²¹ + 201x²⁰ + 133x¹⁹ + 204x¹⁸ + 136x¹⁷ + 200x¹⁶ + 50x¹⁵ + 216x¹⁴

Discard the lead 0 term to get:

60x³¹ + 122x³⁰ + 132x²⁹ + 213x²⁸ + 42x²⁷ + 49x²⁶ + 1x²⁵ + 240x²⁴ + 233x²³ + 102x²² + 126x²¹ + 201x²⁰ + 133x¹⁹ + 204x¹⁸ + 136x¹⁷ + 200x¹⁶ + 50x¹⁵ + 216x¹⁴

Step 2a: Multiply the Generator Polynomial by the Lead Term of the XOR result from the previous step

Next, multiply the generator polynomial by the lead term of the XOR result from the previous step. The lead term in this case is 60x³¹. Convert 60x³¹ to alpha notation. According to the log antilog table, for the integer value 60, the alpha exponent is 77. Therefore 60 = α⁷⁷. Multiply the generator polynomial by α⁷⁷:

(α⁷⁷ * α⁰)x³¹ + (α⁷⁷ * α²¹⁵)x³⁰ + (α⁷⁷ * α²³⁴)x²⁹ + (α⁷⁷ * α¹⁵⁸)x²⁸ + (α⁷⁷ * α⁹⁴)x²⁷ + (α⁷⁷ * α¹⁸⁴)x²⁶ + (α⁷⁷ * α⁹⁷)x²⁵ + (α⁷⁷ * α¹¹⁸)x²⁴ + (α⁷⁷ * α¹⁷⁰)x²³ + (α⁷⁷ * α⁷⁹)x²² + (α⁷⁷ * α¹⁸⁷)x²¹ + (α⁷⁷ * α¹⁵²)x²⁰ + (α⁷⁷ * α¹⁴⁸)x¹⁹ + (α⁷⁷ * α²⁵²)x¹⁸ + (α⁷⁷ * α¹⁷⁹)x¹⁷ + (α⁷⁷ * α⁵)x¹⁶ + (α⁷⁷ * α⁹⁸)x¹⁵ + (α⁷⁷ * α⁹⁶)x¹⁴ + (α⁷⁷ * α¹⁵³)x¹³

The exponents of the alphas are added together. In this case, at least one of the exponents is larger than 255, so perform modulo 255 as follows:

α⁷⁷x³¹ + α^{(292 % 255)}x³⁰ + α^{(311 % 255)}x²⁹ + α²³⁵x²⁸ + α¹⁷¹x²⁷ + α^{(261 % 255)}x²⁶ + α¹⁷⁴x²⁵ + α¹⁹⁵x²⁴ + α²⁴⁷x²³ + α¹⁵⁶x²² + α^{(264 % 255)}x²¹ + α²²⁹x²⁰ + α²²⁵x¹⁹ + α^{(329 % 255)}x¹⁸ + α^{(256 % 255)}x¹⁷ + α⁸²x¹⁶ + α¹⁷⁵x¹⁵ + α¹⁷³x¹⁴ + α²³⁰x¹³

The result is:

α⁷⁷x³¹ + α³⁷x³⁰ + α⁵⁶x²⁹ + α²³⁵x²⁸ + α¹⁷¹x²⁷ + α⁶x²⁶ + α¹⁷⁴x²⁵ + α¹⁹⁵x²⁴ + α²⁴⁷x²³ + α¹⁵⁶x²² + α⁹x²¹ + α²²⁹x²⁰ + α²²⁵x¹⁹ + α⁷⁴x¹⁸ + α¹x¹⁷ + α⁸²x¹⁶ + α¹⁷⁵x¹⁵ + α¹⁷³x¹⁴ + α²³⁰x¹³

Now, convert this to integer notation:

60x³¹ + 74x³⁰ + 93x²⁹ + 235x²⁸ + 179x²⁷ + 64x²⁶ + 241x²⁵ + 100x²⁴ + 131x²³ + 228x²² + 58x²¹ + 122x²⁰ + 36x¹⁹ + 137x¹⁸ + 2x¹⁷ + 211x¹⁶ + 255x¹⁵ + 246x¹⁴ + 244x¹³

Step 2b: XOR the result with the result from step 1b

Use the result from step 1b to perform the next XOR.

(60 ⊕ 60)x³¹ + (122 ⊕ 74)x³⁰ + (132 ⊕ 93)x²⁹ + (213 ⊕ 235)x²⁸ + (42 ⊕ 179)x²⁷ + (49 ⊕ 64)x²⁶ + (1 ⊕ 241)x²⁵ + (240 ⊕ 100)x²⁴ + (233 ⊕ 131)x²³ + (102 ⊕ 228)x²² + (126 ⊕ 58)x²¹ + (201 ⊕ 122)x²⁰ + (133 ⊕ 36)x¹⁹ + (204 ⊕ 137)x¹⁸ + (136 ⊕ 2)x¹⁷ + (200 ⊕ 211)x¹⁶ + (50 ⊕ 255)x¹⁵ + (216 ⊕ 246)x¹⁴ + (0 ⊕ 244)x¹³

The result is:

0x³¹ + 48x³⁰ + 217x²⁹ + 62x²⁸ + 153x²⁷ + 113x²⁶ + 240x²⁵ + 148x²⁴ + 106x²³ + 130x²² + 68x²¹ + 179x²⁰ + 161x¹⁹ + 69x¹⁸ + 138x¹⁷ + 27x¹⁶ + 205x¹⁵ + 46x¹⁴ + 244x¹³

Discard the lead 0 term to get:

48x³⁰ + 217x²⁹ + 62x²⁸ + 153x²⁷ + 113x²⁶ + 240x²⁵ + 148x²⁴ + 106x²³ + 130x²² + 68x²¹ + 179x²⁰ + 161x¹⁹ + 69x¹⁸ + 138x¹⁷ + 27x¹⁶ + 205x¹⁵ + 46x¹⁴ + 244x¹³

Step 3a: Multiply the Generator Polynomial by the Lead Term of the XOR result from the previous step

Next, multiply the generator polynomial by the lead term of the XOR result from the previous step. The lead term in this case is 48x³⁰. Convert 48x³⁰ to alpha notation. According to the log antilog table, for the integer value 48, the alpha exponent is 29. Therefore 48 = α²⁹. Multiply the generator polynomial by α²⁹:

(α²⁹ * α⁰)x³⁰ + (α²⁹ * α²¹⁵)x²⁹ + (α²⁹ * α²³⁴)x²⁸ + (α²⁹ * α¹⁵⁸)x²⁷ + (α²⁹ * α⁹⁴)x²⁶ + (α²⁹ * α¹⁸⁴)x²⁵ + (α²⁹ * α⁹⁷)x²⁴ + (α²⁹ * α¹¹⁸)x²³ + (α²⁹ * α¹⁷⁰)x²² + (α²⁹ * α⁷⁹)x²¹ + (α²⁹ * α¹⁸⁷)x²⁰ + (α²⁹ * α¹⁵²)x¹⁹ + (α²⁹ * α¹⁴⁸)x¹⁸ + (α²⁹ * α²⁵²)x¹⁷ + (α²⁹ * α¹⁷⁹)x¹⁶ + (α²⁹ * α⁵)x¹⁵ + (α²⁹ * α⁹⁸)x¹⁴ + (α²⁹ * α⁹⁶)x¹³ + (α²⁹ * α¹⁵³)x¹²

The exponents of the alphas are added together. In this case, at least one of the exponents is larger than 255, so perform modulo 255 as follows:

α²⁹x³⁰ + α²⁴⁴x²⁹ + α^{(263 % 255)}x²⁸ + α¹⁸⁷x²⁷ + α¹²³x²⁶ + α²¹³x²⁵ + α¹²⁶x²⁴ + α¹⁴⁷x²³ + α¹⁹⁹x²² + α¹⁰⁸x²¹ + α²¹⁶x²⁰ + α¹⁸¹x¹⁹ + α¹⁷⁷x¹⁸ + α^{(281 % 255)}x¹⁷ + α²⁰⁸x¹⁶ + α³⁴x¹⁵ + α¹²⁷x¹⁴ + α¹²⁵x¹³ + α¹⁸²x¹²

The result is:

α²⁹x³⁰ + α²⁴⁴x²⁹ + α⁸x²⁸ + α¹⁸⁷x²⁷ + α¹²³x²⁶ + α²¹³x²⁵ + α¹²⁶x²⁴ + α¹⁴⁷x²³ + α¹⁹⁹x²² + α¹⁰⁸x²¹ + α²¹⁶x²⁰ + α¹⁸¹x¹⁹ + α¹⁷⁷x¹⁸ + α²⁶x¹⁷ + α²⁰⁸x¹⁶ + α³⁴x¹⁵ + α¹²⁷x¹⁴ + α¹²⁵x¹³ + α¹⁸²x¹²

Now, convert this to integer notation:

48x³⁰ + 250x²⁹ + 29x²⁸ + 220x²⁷ + 197x²⁶ + 242x²⁵ + 102x²⁴ + 41x²³ + 14x²² + 208x²¹ + 195x²⁰ + 49x¹⁹ + 219x¹⁸ + 6x¹⁷ + 81x¹⁶ + 78x¹⁵ + 204x¹⁴ + 51x¹³ + 98x¹²

Step 3b: XOR the result with the result from step 2b

Use the result from step 2b to perform the next XOR.

(48 ⊕ 48)x³⁰ + (217 ⊕ 250)x²⁹ + (62 ⊕ 29)x²⁸ + (153 ⊕ 220)x²⁷ + (113 ⊕ 197)x²⁶ + (240 ⊕ 242)x²⁵ + (148 ⊕ 102)x²⁴ + (106 ⊕ 41)x²³ + (130 ⊕ 14)x²² + (68 ⊕ 208)x²¹ + (179 ⊕ 195)x²⁰ + (161 ⊕ 49)x¹⁹ + (69 ⊕ 219)x¹⁸ + (138 ⊕ 6)x¹⁷ + (27 ⊕ 81)x¹⁶ + (205 ⊕ 78)x¹⁵ + (46 ⊕ 204)x¹⁴ + (244 ⊕ 51)x¹³ + (0 ⊕ 98)x¹²

The result is:

0x³⁰ + 35x²⁹ + 35x²⁸ + 69x²⁷ + 180x²⁶ + 2x²⁵ + 242x²⁴ + 67x²³ + 140x²² + 148x²¹ + 112x²⁰ + 144x¹⁹ + 158x¹⁸ + 140x¹⁷ + 74x¹⁶ + 131x¹⁵ + 226x¹⁴ + 199x¹³ + 98x¹²

Discard the lead 0 term to get:

35x²⁹ + 35x²⁸ + 69x²⁷ + 180x²⁶ + 2x²⁵ + 242x²⁴ + 67x²³ + 140x²² + 148x²¹ + 112x²⁰ + 144x¹⁹ + 158x¹⁸ + 140x¹⁷ + 74x¹⁶ + 131x¹⁵ + 226x¹⁴ + 199x¹³ + 98x¹²

Step 4a: Multiply the Generator Polynomial by the Lead Term of the XOR result from the previous step

Next, multiply the generator polynomial by the lead term of the XOR result from the previous step. The lead term in this case is 35x²⁹. Convert 35x²⁹ to alpha notation. According to the log antilog table, for the integer value 35, the alpha exponent is 47. Therefore 35 = α⁴⁷. Multiply the generator polynomial by α⁴⁷:

(α⁴⁷ * α⁰)x²⁹ + (α⁴⁷ * α²¹⁵)x²⁸ + (α⁴⁷ * α²³⁴)x²⁷ + (α⁴⁷ * α¹⁵⁸)x²⁶ + (α⁴⁷ * α⁹⁴)x²⁵ + (α⁴⁷ * α¹⁸⁴)x²⁴ + (α⁴⁷ * α⁹⁷)x²³ + (α⁴⁷ * α¹¹⁸)x²² + (α⁴⁷ * α¹⁷⁰)x²¹ + (α⁴⁷ * α⁷⁹)x²⁰ + (α⁴⁷ * α¹⁸⁷)x¹⁹ + (α⁴⁷ * α¹⁵²)x¹⁸ + (α⁴⁷ * α¹⁴⁸)x¹⁷ + (α⁴⁷ * α²⁵²)x¹⁶ + (α⁴⁷ * α¹⁷⁹)x¹⁵ + (α⁴⁷ * α⁵)x¹⁴ + (α⁴⁷ * α⁹⁸)x¹³ + (α⁴⁷ * α⁹⁶)x¹² + (α⁴⁷ * α¹⁵³)x¹¹

The exponents of the alphas are added together. In this case, at least one of the exponents is larger than 255, so perform modulo 255 as follows:

α⁴⁷x²⁹ + α^{(262 % 255)}x²⁸ + α^{(281 % 255)}x²⁷ + α²⁰⁵x²⁶ + α¹⁴¹x²⁵ + α²³¹x²⁴ + α¹⁴⁴x²³ + α¹⁶⁵x²² + α²¹⁷x²¹ + α¹²⁶x²⁰ + α²³⁴x¹⁹ + α¹⁹⁹x¹⁸ + α¹⁹⁵x¹⁷ + α^{(299 % 255)}x¹⁶ + α²²⁶x¹⁵ + α⁵²x¹⁴ + α¹⁴⁵x¹³ + α¹⁴³x¹² + α²⁰⁰x¹¹

The result is:

α⁴⁷x²⁹ + α⁷x²⁸ + α²⁶x²⁷ + α²⁰⁵x²⁶ + α¹⁴¹x²⁵ + α²³¹x²⁴ + α¹⁴⁴x²³ + α¹⁶⁵x²² + α²¹⁷x²¹ + α¹²⁶x²⁰ + α²³⁴x¹⁹ + α¹⁹⁹x¹⁸ + α¹⁹⁵x¹⁷ + α⁴⁴x¹⁶ + α²²⁶x¹⁵ + α⁵²x¹⁴ + α¹⁴⁵x¹³ + α¹⁴³x¹² + α²⁰⁰x¹¹

Now, convert this to integer notation:

35x²⁹ + 128x²⁸ + 6x²⁷ + 167x²⁶ + 21x²⁵ + 245x²⁴ + 168x²³ + 145x²² + 155x²¹ + 102x²⁰ + 251x¹⁹ + 14x¹⁸ + 100x¹⁷ + 238x¹⁶ + 72x¹⁵ + 20x¹⁴ + 77x¹³ + 84x¹² + 28x¹¹

Step 4b: XOR the result with the result from step 3b

Use the result from step 3b to perform the next XOR.

(35 ⊕ 35)x²⁹ + (35 ⊕ 128)x²⁸ + (69 ⊕ 6)x²⁷ + (180 ⊕ 167)x²⁶ + (2 ⊕ 21)x²⁵ + (242 ⊕ 245)x²⁴ + (67 ⊕ 168)x²³ + (140 ⊕ 145)x²² + (148 ⊕ 155)x²¹ + (112 ⊕ 102)x²⁰ + (144 ⊕ 251)x¹⁹ + (158 ⊕ 14)x¹⁸ + (140 ⊕ 100)x¹⁷ + (74 ⊕ 238)x¹⁶ + (131 ⊕ 72)x¹⁵ + (226 ⊕ 20)x¹⁴ + (199 ⊕ 77)x¹³ + (98 ⊕ 84)x¹² + (0 ⊕ 28)x¹¹

The result is:

0x²⁹ + 163x²⁸ + 67x²⁷ + 19x²⁶ + 23x²⁵ + 7x²⁴ + 235x²³ + 29x²² + 15x²¹ + 22x²⁰ + 107x¹⁹ + 144x¹⁸ + 232x¹⁷ + 164x¹⁶ + 203x¹⁵ + 246x¹⁴ + 138x¹³ + 54x¹² + 28x¹¹

Discard the lead 0 term to get:

163x²⁸ + 67x²⁷ + 19x²⁶ + 23x²⁵ + 7x²⁴ + 235x²³ + 29x²² + 15x²¹ + 22x²⁰ + 107x¹⁹ + 144x¹⁸ + 232x¹⁷ + 164x¹⁶ + 203x¹⁵ + 246x¹⁴ + 138x¹³ + 54x¹² + 28x¹¹

Step 5a: Multiply the Generator Polynomial by the Lead Term of the XOR result from the previous step

Next, multiply the generator polynomial by the lead term of the XOR result from the previous step. The lead term in this case is 163x²⁸. Convert 163x²⁸ to alpha notation. According to the log antilog table, for the integer value 163, the alpha exponent is 91. Therefore 163 = α⁹¹. Multiply the generator polynomial by α⁹¹:

(α⁹¹ * α⁰)x²⁸ + (α⁹¹ * α²¹⁵)x²⁷ + (α⁹¹ * α²³⁴)x²⁶ + (α⁹¹ * α¹⁵⁸)x²⁵ + (α⁹¹ * α⁹⁴)x²⁴ + (α⁹¹ * α¹⁸⁴)x²³ + (α⁹¹ * α⁹⁷)x²² + (α⁹¹ * α¹¹⁸)x²¹ + (α⁹¹ * α¹⁷⁰)x²⁰ + (α⁹¹ * α⁷⁹)x¹⁹ + (α⁹¹ * α¹⁸⁷)x¹⁸ + (α⁹¹ * α¹⁵²)x¹⁷ + (α⁹¹ * α¹⁴⁸)x¹⁶ + (α⁹¹ * α²⁵²)x¹⁵ + (α⁹¹ * α¹⁷⁹)x¹⁴ + (α⁹¹ * α⁵)x¹³ + (α⁹¹ * α⁹⁸)x¹² + (α⁹¹ * α⁹⁶)x¹¹ + (α⁹¹ * α¹⁵³)x¹⁰

The exponents of the alphas are added together. In this case, at least one of the exponents is larger than 255, so perform modulo 255 as follows:

α⁹¹x²⁸ + α^{(306 % 255)}x²⁷ + α^{(325 % 255)}x²⁶ + α²⁴⁹x²⁵ + α¹⁸⁵x²⁴ + α^{(275 % 255)}x²³ + α¹⁸⁸x²² + α²⁰⁹x²¹ + α^{(261 % 255)}x²⁰ + α¹⁷⁰x¹⁹ + α^{(278 % 255)}x¹⁸ + α²⁴³x¹⁷ + α²³⁹x¹⁶ + α^{(343 % 255)}x¹⁵ + α^{(270 % 255)}x¹⁴ + α⁹⁶x¹³ + α¹⁸⁹x¹² + α¹⁸⁷x¹¹ + α²⁴⁴x¹⁰

The result is:

α⁹¹x²⁸ + α⁵¹x²⁷ + α⁷⁰x²⁶ + α²⁴⁹x²⁵ + α¹⁸⁵x²⁴ + α²⁰x²³ + α¹⁸⁸x²² + α²⁰⁹x²¹ + α⁶x²⁰ + α¹⁷⁰x¹⁹ + α²³x¹⁸ + α²⁴³x¹⁷ + α²³⁹x¹⁶ + α⁸⁸x¹⁵ + α¹⁵x¹⁴ + α⁹⁶x¹³ + α¹⁸⁹x¹² + α¹⁸⁷x¹¹ + α²⁴⁴x¹⁰

Now, convert this to integer notation:

163x²⁸ + 10x²⁷ + 94x²⁶ + 54x²⁵ + 55x²⁴ + 180x²³ + 165x²² + 162x²¹ + 64x²⁰ + 215x¹⁹ + 201x¹⁸ + 125x¹⁷ + 22x¹⁶ + 254x¹⁵ + 38x¹⁴ + 217x¹³ + 87x¹² + 220x¹¹ + 250x¹⁰

Step 5b: XOR the result with the result from step 4b

Use the result from step 4b to perform the next XOR.

(163 ⊕ 163)x²⁸ + (67 ⊕ 10)x²⁷ + (19 ⊕ 94)x²⁶ + (23 ⊕ 54)x²⁵ + (7 ⊕ 55)x²⁴ + (235 ⊕ 180)x²³ + (29 ⊕ 165)x²² + (15 ⊕ 162)x²¹ + (22 ⊕ 64)x²⁰ + (107 ⊕ 215)x¹⁹ + (144 ⊕ 201)x¹⁸ + (232 ⊕ 125)x¹⁷ + (164 ⊕ 22)x¹⁶ + (203 ⊕ 254)x¹⁵ + (246 ⊕ 38)x¹⁴ + (138 ⊕ 217)x¹³ + (54 ⊕ 87)x¹² + (28 ⊕ 220)x¹¹ + (0 ⊕ 250)x¹⁰

The result is:

0x²⁸ + 73x²⁷ + 77x²⁶ + 33x²⁵ + 48x²⁴ + 95x²³ + 184x²² + 173x²¹ + 86x²⁰ + 188x¹⁹ + 89x¹⁸ + 149x¹⁷ + 178x¹⁶ + 53x¹⁵ + 208x¹⁴ + 83x¹³ + 97x¹² + 192x¹¹ + 250x¹⁰

Discard the lead 0 term to get:

73x²⁷ + 77x²⁶ + 33x²⁵ + 48x²⁴ + 95x²³ + 184x²² + 173x²¹ + 86x²⁰ + 188x¹⁹ + 89x¹⁸ + 149x¹⁷ + 178x¹⁶ + 53x¹⁵ + 208x¹⁴ + 83x¹³ + 97x¹² + 192x¹¹ + 250x¹⁰

Step 6a: Multiply the Generator Polynomial by the Lead Term of the XOR result from the previous step

Next, multiply the generator polynomial by the lead term of the XOR result from the previous step. The lead term in this case is 73x²⁷. Convert 73x²⁷ to alpha notation. According to the log antilog table, for the integer value 73, the alpha exponent is 152. Therefore 73 = α¹⁵². Multiply the generator polynomial by α¹⁵²:

(α¹⁵² * α⁰)x²⁷ + (α¹⁵² * α²¹⁵)x²⁶ + (α¹⁵² * α²³⁴)x²⁵ + (α¹⁵² * α¹⁵⁸)x²⁴ + (α¹⁵² * α⁹⁴)x²³ + (α¹⁵² * α¹⁸⁴)x²² + (α¹⁵² * α⁹⁷)x²¹ + (α¹⁵² * α¹¹⁸)x²⁰ + (α¹⁵² * α¹⁷⁰)x¹⁹ + (α¹⁵² * α⁷⁹)x¹⁸ + (α¹⁵² * α¹⁸⁷)x¹⁷ + (α¹⁵² * α¹⁵²)x¹⁶ + (α¹⁵² * α¹⁴⁸)x¹⁵ + (α¹⁵² * α²⁵²)x¹⁴ + (α¹⁵² * α¹⁷⁹)x¹³ + (α¹⁵² * α⁵)x¹² + (α¹⁵² * α⁹⁸)x¹¹ + (α¹⁵² * α⁹⁶)x¹⁰ + (α¹⁵² * α¹⁵³)x⁹

The exponents of the alphas are added together. In this case, at least one of the exponents is larger than 255, so perform modulo 255 as follows:

α¹⁵²x²⁷ + α^{(367 % 255)}x²⁶ + α^{(386 % 255)}x²⁵ + α^{(310 % 255)}x²⁴ + α²⁴⁶x²³ + α^{(336 % 255)}x²² + α²⁴⁹x²¹ + α^{(270 % 255)}x²⁰ + α^{(322 % 255)}x¹⁹ + α²³¹x¹⁸ + α^{(339 % 255)}x¹⁷ + α^{(304 % 255)}x¹⁶ + α^{(300 % 255)}x¹⁵ + α^{(404 % 255)}x¹⁴ + α^{(331 % 255)}x¹³ + α¹⁵⁷x¹² + α²⁵⁰x¹¹ + α²⁴⁸x¹⁰ + α^{(305 % 255)}x⁹

The result is:

α¹⁵²x²⁷ + α¹¹²x²⁶ + α¹³¹x²⁵ + α⁵⁵x²⁴ + α²⁴⁶x²³ + α⁸¹x²² + α²⁴⁹x²¹ + α¹⁵x²⁰ + α⁶⁷x¹⁹ + α²³¹x¹⁸ + α⁸⁴x¹⁷ + α⁴⁹x¹⁶ + α⁴⁵x¹⁵ + α¹⁴⁹x¹⁴ + α⁷⁶x¹³ + α¹⁵⁷x¹² + α²⁵⁰x¹¹ + α²⁴⁸x¹⁰ + α⁵⁰x⁹

Now, convert this to integer notation:

73x²⁷ + 129x²⁶ + 92x²⁵ + 160x²⁴ + 207x²³ + 231x²² + 54x²¹ + 38x²⁰ + 194x¹⁹ + 245x¹⁸ + 107x¹⁷ + 140x¹⁶ + 193x¹⁵ + 164x¹⁴ + 30x¹³ + 213x¹² + 108x¹¹ + 27x¹⁰ + 5x⁹

Step 6b: XOR the result with the result from step 5b

Use the result from step 5b to perform the next XOR.

(73 ⊕ 73)x²⁷ + (77 ⊕ 129)x²⁶ + (33 ⊕ 92)x²⁵ + (48 ⊕ 160)x²⁴ + (95 ⊕ 207)x²³ + (184 ⊕ 231)x²² + (173 ⊕ 54)x²¹ + (86 ⊕ 38)x²⁰ + (188 ⊕ 194)x¹⁹ + (89 ⊕ 245)x¹⁸ + (149 ⊕ 107)x¹⁷ + (178 ⊕ 140)x¹⁶ + (53 ⊕ 193)x¹⁵ + (208 ⊕ 164)x¹⁴ + (83 ⊕ 30)x¹³ + (97 ⊕ 213)x¹² + (192 ⊕ 108)x¹¹ + (250 ⊕ 27)x¹⁰ + (0 ⊕ 5)x⁹

The result is:

0x²⁷ + 204x²⁶ + 125x²⁵ + 144x²⁴ + 144x²³ + 95x²² + 155x²¹ + 112x²⁰ + 126x¹⁹ + 172x¹⁸ + 254x¹⁷ + 62x¹⁶ + 244x¹⁵ + 116x¹⁴ + 77x¹³ + 180x¹² + 172x¹¹ + 225x¹⁰ + 5x⁹

Discard the lead 0 term to get:

204x²⁶ + 125x²⁵ + 144x²⁴ + 144x²³ + 95x²² + 155x²¹ + 112x²⁰ + 126x¹⁹ + 172x¹⁸ + 254x¹⁷ + 62x¹⁶ + 244x¹⁵ + 116x¹⁴ + 77x¹³ + 180x¹² + 172x¹¹ + 225x¹⁰ + 5x⁹

Step 7a: Multiply the Generator Polynomial by the Lead Term of the XOR result from the previous step

Next, multiply the generator polynomial by the lead term of the XOR result from the previous step. The lead term in this case is 204x²⁶. Convert 204x²⁶ to alpha notation. According to the log antilog table, for the integer value 204, the alpha exponent is 127. Therefore 204 = α¹²⁷. Multiply the generator polynomial by α¹²⁷:

(α¹²⁷ * α⁰)x²⁶ + (α¹²⁷ * α²¹⁵)x²⁵ + (α¹²⁷ * α²³⁴)x²⁴ + (α¹²⁷ * α¹⁵⁸)x²³ + (α¹²⁷ * α⁹⁴)x²² + (α¹²⁷ * α¹⁸⁴)x²¹ + (α¹²⁷ * α⁹⁷)x²⁰ + (α¹²⁷ * α¹¹⁸)x¹⁹ + (α¹²⁷ * α¹⁷⁰)x¹⁸ + (α¹²⁷ * α⁷⁹)x¹⁷ + (α¹²⁷ * α¹⁸⁷)x¹⁶ + (α¹²⁷ * α¹⁵²)x¹⁵ + (α¹²⁷ * α¹⁴⁸)x¹⁴ + (α¹²⁷ * α²⁵²)x¹³ + (α¹²⁷ * α¹⁷⁹)x¹² + (α¹²⁷ * α⁵)x¹¹ + (α¹²⁷ * α⁹⁸)x¹⁰ + (α¹²⁷ * α⁹⁶)x⁹ + (α¹²⁷ * α¹⁵³)x⁸

The exponents of the alphas are added together. In this case, at least one of the exponents is larger than 255, so perform modulo 255 as follows:

α¹²⁷x²⁶ + α^{(342 % 255)}x²⁵ + α^{(361 % 255)}x²⁴ + α^{(285 % 255)}x²³ + α²²¹x²² + α^{(311 % 255)}x²¹ + α²²⁴x²⁰ + α²⁴⁵x¹⁹ + α^{(297 % 255)}x¹⁸ + α²⁰⁶x¹⁷ + α^{(314 % 255)}x¹⁶ + α^{(279 % 255)}x¹⁵ + α^{(275 % 255)}x¹⁴ + α^{(379 % 255)}x¹³ + α^{(306 % 255)}x¹² + α¹³²x¹¹ + α²²⁵x¹⁰ + α²²³x⁹ + α^{(280 % 255)}x⁸

The result is:

α¹²⁷x²⁶ + α⁸⁷x²⁵ + α¹⁰⁶x²⁴ + α³⁰x²³ + α²²¹x²² + α⁵⁶x²¹ + α²²⁴x²⁰ + α²⁴⁵x¹⁹ + α⁴²x¹⁸ + α²⁰⁶x¹⁷ + α⁵⁹x¹⁶ + α²⁴x¹⁵ + α²⁰x¹⁴ + α¹²⁴x¹³ + α⁵¹x¹² + α¹³²x¹¹ + α²²⁵x¹⁰ + α²²³x⁹ + α²⁵x⁸

Now, convert this to integer notation:

204x²⁶ + 127x²⁵ + 52x²⁴ + 96x²³ + 69x²² + 93x²¹ + 18x²⁰ + 233x¹⁹ + 181x¹⁸ + 83x¹⁷ + 210x¹⁶ + 143x¹⁵ + 180x¹⁴ + 151x¹³ + 10x¹² + 184x¹¹ + 36x¹⁰ + 9x⁹ + 3x⁸

Step 7b: XOR the result with the result from step 6b

Use the result from step 6b to perform the next XOR.

(204 ⊕ 204)x²⁶ + (125 ⊕ 127)x²⁵ + (144 ⊕ 52)x²⁴ + (144 ⊕ 96)x²³ + (95 ⊕ 69)x²² + (155 ⊕ 93)x²¹ + (112 ⊕ 18)x²⁰ + (126 ⊕ 233)x¹⁹ + (172 ⊕ 181)x¹⁸ + (254 ⊕ 83)x¹⁷ + (62 ⊕ 210)x¹⁶ + (244 ⊕ 143)x¹⁵ + (116 ⊕ 180)x¹⁴ + (77 ⊕ 151)x¹³ + (180 ⊕ 10)x¹² + (172 ⊕ 184)x¹¹ + (225 ⊕ 36)x¹⁰ + (5 ⊕ 9)x⁹ + (0 ⊕ 3)x⁸

The result is:

0x²⁶ + 2x²⁵ + 164x²⁴ + 240x²³ + 26x²² + 198x²¹ + 98x²⁰ + 151x¹⁹ + 25x¹⁸ + 173x¹⁷ + 236x¹⁶ + 123x¹⁵ + 192x¹⁴ + 218x¹³ + 190x¹² + 20x¹¹ + 197x¹⁰ + 12x⁹ + 3x⁸

Discard the lead 0 term to get:

2x²⁵ + 164x²⁴ + 240x²³ + 26x²² + 198x²¹ + 98x²⁰ + 151x¹⁹ + 25x¹⁸ + 173x¹⁷ + 236x¹⁶ + 123x¹⁵ + 192x¹⁴ + 218x¹³ + 190x¹² + 20x¹¹ + 197x¹⁰ + 12x⁹ + 3x⁸

Step 8a: Multiply the Generator Polynomial by the Lead Term of the XOR result from the previous step

Next, multiply the generator polynomial by the lead term of the XOR result from the previous step. The lead term in this case is 2x²⁵. Convert 2x²⁵ to alpha notation. According to the log antilog table, for the integer value 2, the alpha exponent is 1. Therefore 2 = α¹. Multiply the generator polynomial by α¹:

(α¹ * α⁰)x²⁵ + (α¹ * α²¹⁵)x²⁴ + (α¹ * α²³⁴)x²³ + (α¹ * α¹⁵⁸)x²² + (α¹ * α⁹⁴)x²¹ + (α¹ * α¹⁸⁴)x²⁰ + (α¹ * α⁹⁷)x¹⁹ + (α¹ * α¹¹⁸)x¹⁸ + (α¹ * α¹⁷⁰)x¹⁷ + (α¹ * α⁷⁹)x¹⁶ + (α¹ * α¹⁸⁷)x¹⁵ + (α¹ * α¹⁵²)x¹⁴ + (α¹ * α¹⁴⁸)x¹³ + (α¹ * α²⁵²)x¹² + (α¹ * α¹⁷⁹)x¹¹ + (α¹ * α⁵)x¹⁰ + (α¹ * α⁹⁸)x⁹ + (α¹ * α⁹⁶)x⁸ + (α¹ * α¹⁵³)x⁷

The exponents of the alphas are added together. The result is:

α¹x²⁵ + α²¹⁶x²⁴ + α²³⁵x²³ + α¹⁵⁹x²² + α⁹⁵x²¹ + α¹⁸⁵x²⁰ + α⁹⁸x¹⁹ + α¹¹⁹x¹⁸ + α¹⁷¹x¹⁷ + α⁸⁰x¹⁶ + α¹⁸⁸x¹⁵ + α¹⁵³x¹⁴ + α¹⁴⁹x¹³ + α²⁵³x¹² + α¹⁸⁰x¹¹ + α⁶x¹⁰ + α⁹⁹x⁹ + α⁹⁷x⁸ + α¹⁵⁴x⁷

Now, convert this to integer notation:

2x²⁵ + 195x²⁴ + 235x²³ + 115x²² + 226x²¹ + 55x²⁰ + 67x¹⁹ + 147x¹⁸ + 179x¹⁷ + 253x¹⁶ + 165x¹⁵ + 146x¹⁴ + 164x¹³ + 71x¹² + 150x¹¹ + 64x¹⁰ + 134x⁹ + 175x⁸ + 57x⁷

Step 8b: XOR the result with the result from step 7b

Use the result from step 7b to perform the next XOR.

(2 ⊕ 2)x²⁵ + (164 ⊕ 195)x²⁴ + (240 ⊕ 235)x²³ + (26 ⊕ 115)x²² + (198 ⊕ 226)x²¹ + (98 ⊕ 55)x²⁰ + (151 ⊕ 67)x¹⁹ + (25 ⊕ 147)x¹⁸ + (173 ⊕ 179)x¹⁷ + (236 ⊕ 253)x¹⁶ + (123 ⊕ 165)x¹⁵ + (192 ⊕ 146)x¹⁴ + (218 ⊕ 164)x¹³ + (190 ⊕ 71)x¹² + (20 ⊕ 150)x¹¹ + (197 ⊕ 64)x¹⁰ + (12 ⊕ 134)x⁹ + (3 ⊕ 175)x⁸ + (0 ⊕ 57)x⁷

The result is:

0x²⁵ + 103x²⁴ + 27x²³ + 105x²² + 36x²¹ + 85x²⁰ + 212x¹⁹ + 138x¹⁸ + 30x¹⁷ + 17x¹⁶ + 222x¹⁵ + 82x¹⁴ + 126x¹³ + 249x¹² + 130x¹¹ + 133x¹⁰ + 138x⁹ + 172x⁸ + 57x⁷

Discard the lead 0 term to get:

103x²⁴ + 27x²³ + 105x²² + 36x²¹ + 85x²⁰ + 212x¹⁹ + 138x¹⁸ + 30x¹⁷ + 17x¹⁶ + 222x¹⁵ + 82x¹⁴ + 126x¹³ + 249x¹² + 130x¹¹ + 133x¹⁰ + 138x⁹ + 172x⁸ + 57x⁷

Step 9a: Multiply the Generator Polynomial by the Lead Term of the XOR result from the previous step

Next, multiply the generator polynomial by the lead term of the XOR result from the previous step. The lead term in this case is 103x²⁴. Convert 103x²⁴ to alpha notation. According to the log antilog table, for the integer value 103, the alpha exponent is 110. Therefore 103 = α¹¹⁰. Multiply the generator polynomial by α¹¹⁰:

(α¹¹⁰ * α⁰)x²⁴ + (α¹¹⁰ * α²¹⁵)x²³ + (α¹¹⁰ * α²³⁴)x²² + (α¹¹⁰ * α¹⁵⁸)x²¹ + (α¹¹⁰ * α⁹⁴)x²⁰ + (α¹¹⁰ * α¹⁸⁴)x¹⁹ + (α¹¹⁰ * α⁹⁷)x¹⁸ + (α¹¹⁰ * α¹¹⁸)x¹⁷ + (α¹¹⁰ * α¹⁷⁰)x¹⁶ + (α¹¹⁰ * α⁷⁹)x¹⁵ + (α¹¹⁰ * α¹⁸⁷)x¹⁴ + (α¹¹⁰ * α¹⁵²)x¹³ + (α¹¹⁰ * α¹⁴⁸)x¹² + (α¹¹⁰ * α²⁵²)x¹¹ + (α¹¹⁰ * α¹⁷⁹)x¹⁰ + (α¹¹⁰ * α⁵)x⁹ + (α¹¹⁰ * α⁹⁸)x⁸ + (α¹¹⁰ * α⁹⁶)x⁷ + (α¹¹⁰ * α¹⁵³)x⁶

The exponents of the alphas are added together. In this case, at least one of the exponents is larger than 255, so perform modulo 255 as follows:

α¹¹⁰x²⁴ + α^{(325 % 255)}x²³ + α^{(344 % 255)}x²² + α^{(268 % 255)}x²¹ + α²⁰⁴x²⁰ + α^{(294 % 255)}x¹⁹ + α²⁰⁷x¹⁸ + α²²⁸x¹⁷ + α^{(280 % 255)}x¹⁶ + α¹⁸⁹x¹⁵ + α^{(297 % 255)}x¹⁴ + α^{(262 % 255)}x¹³ + α^{(258 % 255)}x¹² + α^{(362 % 255)}x¹¹ + α^{(289 % 255)}x¹⁰ + α¹¹⁵x⁹ + α²⁰⁸x⁸ + α²⁰⁶x⁷ + α^{(263 % 255)}x⁶

The result is:

α¹¹⁰x²⁴ + α⁷⁰x²³ + α⁸⁹x²² + α¹³x²¹ + α²⁰⁴x²⁰ + α³⁹x¹⁹ + α²⁰⁷x¹⁸ + α²²⁸x¹⁷ + α²⁵x¹⁶ + α¹⁸⁹x¹⁵ + α⁴²x¹⁴ + α⁷x¹³ + α³x¹² + α¹⁰⁷x¹¹ + α³⁴x¹⁰ + α¹¹⁵x⁹ + α²⁰⁸x⁸ + α²⁰⁶x⁷ + α⁸x⁶

Now, convert this to integer notation:

103x²⁴ + 94x²³ + 225x²² + 135x²¹ + 221x²⁰ + 53x¹⁹ + 166x¹⁸ + 61x¹⁷ + 3x¹⁶ + 87x¹⁵ + 181x¹⁴ + 128x¹³ + 8x¹² + 104x¹¹ + 78x¹⁰ + 124x⁹ + 81x⁸ + 83x⁷ + 29x⁶

Step 9b: XOR the result with the result from step 8b

Use the result from step 8b to perform the next XOR.

(103 ⊕ 103)x²⁴ + (27 ⊕ 94)x²³ + (105 ⊕ 225)x²² + (36 ⊕ 135)x²¹ + (85 ⊕ 221)x²⁰ + (212 ⊕ 53)x¹⁹ + (138 ⊕ 166)x¹⁸ + (30 ⊕ 61)x¹⁷ + (17 ⊕ 3)x¹⁶ + (222 ⊕ 87)x¹⁵ + (82 ⊕ 181)x¹⁴ + (126 ⊕ 128)x¹³ + (249 ⊕ 8)x¹² + (130 ⊕ 104)x¹¹ + (133 ⊕ 78)x¹⁰ + (138 ⊕ 124)x⁹ + (172 ⊕ 81)x⁸ + (57 ⊕ 83)x⁷ + (0 ⊕ 29)x⁶

The result is:

0x²⁴ + 69x²³ + 136x²² + 163x²¹ + 136x²⁰ + 225x¹⁹ + 44x¹⁸ + 35x¹⁷ + 18x¹⁶ + 137x¹⁵ + 231x¹⁴ + 254x¹³ + 241x¹² + 234x¹¹ + 203x¹⁰ + 246x⁹ + 253x⁸ + 106x⁷ + 29x⁶

Discard the lead 0 term to get:

69x²³ + 136x²² + 163x²¹ + 136x²⁰ + 225x¹⁹ + 44x¹⁸ + 35x¹⁷ + 18x¹⁶ + 137x¹⁵ + 231x¹⁴ + 254x¹³ + 241x¹² + 234x¹¹ + 203x¹⁰ + 246x⁹ + 253x⁸ + 106x⁷ + 29x⁶

Step 10a: Multiply the Generator Polynomial by the Lead Term of the XOR result from the previous step

Next, multiply the generator polynomial by the lead term of the XOR result from the previous step. The lead term in this case is 69x²³. Convert 69x²³ to alpha notation. According to the log antilog table, for the integer value 69, the alpha exponent is 221. Therefore 69 = α²²¹. Multiply the generator polynomial by α²²¹:

(α²²¹ * α⁰)x²³ + (α²²¹ * α²¹⁵)x²² + (α²²¹ * α²³⁴)x²¹ + (α²²¹ * α¹⁵⁸)x²⁰ + (α²²¹ * α⁹⁴)x¹⁹ + (α²²¹ * α¹⁸⁴)x¹⁸ + (α²²¹ * α⁹⁷)x¹⁷ + (α²²¹ * α¹¹⁸)x¹⁶ + (α²²¹ * α¹⁷⁰)x¹⁵ + (α²²¹ * α⁷⁹)x¹⁴ + (α²²¹ * α¹⁸⁷)x¹³ + (α²²¹ * α¹⁵²)x¹² + (α²²¹ * α¹⁴⁸)x¹¹ + (α²²¹ * α²⁵²)x¹⁰ + (α²²¹ * α¹⁷⁹)x⁹ + (α²²¹ * α⁵)x⁸ + (α²²¹ * α⁹⁸)x⁷ + (α²²¹ * α⁹⁶)x⁶ + (α²²¹ * α¹⁵³)x⁵

The exponents of the alphas are added together. In this case, at least one of the exponents is larger than 255, so perform modulo 255 as follows:

α²²¹x²³ + α^{(436 % 255)}x²² + α^{(455 % 255)}x²¹ + α^{(379 % 255)}x²⁰ + α^{(315 % 255)}x¹⁹ + α^{(405 % 255)}x¹⁸ + α^{(318 % 255)}x¹⁷ + α^{(339 % 255)}x¹⁶ + α^{(391 % 255)}x¹⁵ + α^{(300 % 255)}x¹⁴ + α^{(408 % 255)}x¹³ + α^{(373 % 255)}x¹² + α^{(369 % 255)}x¹¹ + α^{(473 % 255)}x¹⁰ + α^{(400 % 255)}x⁹ + α²²⁶x⁸ + α^{(319 % 255)}x⁷ + α^{(317 % 255)}x⁶ + α^{(374 % 255)}x⁵

The result is:

α²²¹x²³ + α¹⁸¹x²² + α²⁰⁰x²¹ + α¹²⁴x²⁰ + α⁶⁰x¹⁹ + α¹⁵⁰x¹⁸ + α⁶³x¹⁷ + α⁸⁴x¹⁶ + α¹³⁶x¹⁵ + α⁴⁵x¹⁴ + α¹⁵³x¹³ + α¹¹⁸x¹² + α¹¹⁴x¹¹ + α²¹⁸x¹⁰ + α¹⁴⁵x⁹ + α²²⁶x⁸ + α⁶⁴x⁷ + α⁶²x⁶ + α¹¹⁹x⁵

Now, convert this to integer notation:

69x²³ + 49x²² + 28x²¹ + 151x²⁰ + 185x¹⁹ + 85x¹⁸ + 161x¹⁷ + 107x¹⁶ + 79x¹⁵ + 193x¹⁴ + 146x¹³ + 199x¹² + 62x¹¹ + 43x¹⁰ + 77x⁹ + 72x⁸ + 95x⁷ + 222x⁶ + 147x⁵

Step 10b: XOR the result with the result from step 9b

Use the result from step 9b to perform the next XOR.

(69 ⊕ 69)x²³ + (136 ⊕ 49)x²² + (163 ⊕ 28)x²¹ + (136 ⊕ 151)x²⁰ + (225 ⊕ 185)x¹⁹ + (44 ⊕ 85)x¹⁸ + (35 ⊕ 161)x¹⁷ + (18 ⊕ 107)x¹⁶ + (137 ⊕ 79)x¹⁵ + (231 ⊕ 193)x¹⁴ + (254 ⊕ 146)x¹³ + (241 ⊕ 199)x¹² + (234 ⊕ 62)x¹¹ + (203 ⊕ 43)x¹⁰ + (246 ⊕ 77)x⁹ + (253 ⊕ 72)x⁸ + (106 ⊕ 95)x⁷ + (29 ⊕ 222)x⁶ + (0 ⊕ 147)x⁵

The result is:

0x²³ + 185x²² + 191x²¹ + 31x²⁰ + 88x¹⁹ + 121x¹⁸ + 130x¹⁷ + 121x¹⁶ + 198x¹⁵ + 38x¹⁴ + 108x¹³ + 54x¹² + 212x¹¹ + 224x¹⁰ + 187x⁹ + 181x⁸ + 53x⁷ + 195x⁶ + 147x⁵

Discard the lead 0 term to get:

185x²² + 191x²¹ + 31x²⁰ + 88x¹⁹ + 121x¹⁸ + 130x¹⁷ + 121x¹⁶ + 198x¹⁵ + 38x¹⁴ + 108x¹³ + 54x¹² + 212x¹¹ + 224x¹⁰ + 187x⁹ + 181x⁸ + 53x⁷ + 195x⁶ + 147x⁵

Step 11a: Multiply the Generator Polynomial by the Lead Term of the XOR result from the previous step

Next, multiply the generator polynomial by the lead term of the XOR result from the previous step. The lead term in this case is 185x²². Convert 185x²² to alpha notation. According to the log antilog table, for the integer value 185, the alpha exponent is 60. Therefore 185 = α⁶⁰. Multiply the generator polynomial by α⁶⁰:

(α⁶⁰ * α⁰)x²² + (α⁶⁰ * α²¹⁵)x²¹ + (α⁶⁰ * α²³⁴)x²⁰ + (α⁶⁰ * α¹⁵⁸)x¹⁹ + (α⁶⁰ * α⁹⁴)x¹⁸ + (α⁶⁰ * α¹⁸⁴)x¹⁷ + (α⁶⁰ * α⁹⁷)x¹⁶ + (α⁶⁰ * α¹¹⁸)x¹⁵ + (α⁶⁰ * α¹⁷⁰)x¹⁴ + (α⁶⁰ * α⁷⁹)x¹³ + (α⁶⁰ * α¹⁸⁷)x¹² + (α⁶⁰ * α¹⁵²)x¹¹ + (α⁶⁰ * α¹⁴⁸)x¹⁰ + (α⁶⁰ * α²⁵²)x⁹ + (α⁶⁰ * α¹⁷⁹)x⁸ + (α⁶⁰ * α⁵)x⁷ + (α⁶⁰ * α⁹⁸)x⁶ + (α⁶⁰ * α⁹⁶)x⁵ + (α⁶⁰ * α¹⁵³)x⁴

The exponents of the alphas are added together. In this case, at least one of the exponents is larger than 255, so perform modulo 255 as follows:

α⁶⁰x²² + α^{(275 % 255)}x²¹ + α^{(294 % 255)}x²⁰ + α²¹⁸x¹⁹ + α¹⁵⁴x¹⁸ + α²⁴⁴x¹⁷ + α¹⁵⁷x¹⁶ + α¹⁷⁸x¹⁵ + α²³⁰x¹⁴ + α¹³⁹x¹³ + α²⁴⁷x¹² + α²¹²x¹¹ + α²⁰⁸x¹⁰ + α^{(312 % 255)}x⁹ + α²³⁹x⁸ + α⁶⁵x⁷ + α¹⁵⁸x⁶ + α¹⁵⁶x⁵ + α²¹³x⁴

The result is:

α⁶⁰x²² + α²⁰x²¹ + α³⁹x²⁰ + α²¹⁸x¹⁹ + α¹⁵⁴x¹⁸ + α²⁴⁴x¹⁷ + α¹⁵⁷x¹⁶ + α¹⁷⁸x¹⁵ + α²³⁰x¹⁴ + α¹³⁹x¹³ + α²⁴⁷x¹² + α²¹²x¹¹ + α²⁰⁸x¹⁰ + α⁵⁷x⁹ + α²³⁹x⁸ + α⁶⁵x⁷ + α¹⁵⁸x⁶ + α¹⁵⁶x⁵ + α²¹³x⁴

Now, convert this to integer notation:

185x²² + 180x²¹ + 53x²⁰ + 43x¹⁹ + 57x¹⁸ + 250x¹⁷ + 213x¹⁶ + 171x¹⁵ + 244x¹⁴ + 66x¹³ + 131x¹² + 121x¹¹ + 81x¹⁰ + 186x⁹ + 22x⁸ + 190x⁷ + 183x⁶ + 228x⁵ + 242x⁴

Step 11b: XOR the result with the result from step 10b

Use the result from step 10b to perform the next XOR.

(185 ⊕ 185)x²² + (191 ⊕ 180)x²¹ + (31 ⊕ 53)x²⁰ + (88 ⊕ 43)x¹⁹ + (121 ⊕ 57)x¹⁸ + (130 ⊕ 250)x¹⁷ + (121 ⊕ 213)x¹⁶ + (198 ⊕ 171)x¹⁵ + (38 ⊕ 244)x¹⁴ + (108 ⊕ 66)x¹³ + (54 ⊕ 131)x¹² + (212 ⊕ 121)x¹¹ + (224 ⊕ 81)x¹⁰ + (187 ⊕ 186)x⁹ + (181 ⊕ 22)x⁸ + (53 ⊕ 190)x⁷ + (195 ⊕ 183)x⁶ + (147 ⊕ 228)x⁵ + (0 ⊕ 242)x⁴

The result is:

0x²² + 11x²¹ + 42x²⁰ + 115x¹⁹ + 64x¹⁸ + 120x¹⁷ + 172x¹⁶ + 109x¹⁵ + 210x¹⁴ + 46x¹³ + 181x¹² + 173x¹¹ + 177x¹⁰ + 1x⁹ + 163x⁸ + 139x⁷ + 116x⁶ + 119x⁵ + 242x⁴

Discard the lead 0 term to get:

11x²¹ + 42x²⁰ + 115x¹⁹ + 64x¹⁸ + 120x¹⁷ + 172x¹⁶ + 109x¹⁵ + 210x¹⁴ + 46x¹³ + 181x¹² + 173x¹¹ + 177x¹⁰ + 1x⁹ + 163x⁸ + 139x⁷ + 116x⁶ + 119x⁵ + 242x⁴

Step 12a: Multiply the Generator Polynomial by the Lead Term of the XOR result from the previous step

Next, multiply the generator polynomial by the lead term of the XOR result from the previous step. The lead term in this case is 11x²¹. Convert 11x²¹ to alpha notation. According to the log antilog table, for the integer value 11, the alpha exponent is 238. Therefore 11 = α²³⁸. Multiply the generator polynomial by α²³⁸:

(α²³⁸ * α⁰)x²¹ + (α²³⁸ * α²¹⁵)x²⁰ + (α²³⁸ * α²³⁴)x¹⁹ + (α²³⁸ * α¹⁵⁸)x¹⁸ + (α²³⁸ * α⁹⁴)x¹⁷ + (α²³⁸ * α¹⁸⁴)x¹⁶ + (α²³⁸ * α⁹⁷)x¹⁵ + (α²³⁸ * α¹¹⁸)x¹⁴ + (α²³⁸ * α¹⁷⁰)x¹³ + (α²³⁸ * α⁷⁹)x¹² + (α²³⁸ * α¹⁸⁷)x¹¹ + (α²³⁸ * α¹⁵²)x¹⁰ + (α²³⁸ * α¹⁴⁸)x⁹ + (α²³⁸ * α²⁵²)x⁸ + (α²³⁸ * α¹⁷⁹)x⁷ + (α²³⁸ * α⁵)x⁶ + (α²³⁸ * α⁹⁸)x⁵ + (α²³⁸ * α⁹⁶)x⁴ + (α²³⁸ * α¹⁵³)x³

The exponents of the alphas are added together. In this case, at least one of the exponents is larger than 255, so perform modulo 255 as follows:

α²³⁸x²¹ + α^{(453 % 255)}x²⁰ + α^{(472 % 255)}x¹⁹ + α^{(396 % 255)}x¹⁸ + α^{(332 % 255)}x¹⁷ + α^{(422 % 255)}x¹⁶ + α^{(335 % 255)}x¹⁵ + α^{(356 % 255)}x¹⁴ + α^{(408 % 255)}x¹³ + α^{(317 % 255)}x¹² + α^{(425 % 255)}x¹¹ + α^{(390 % 255)}x¹⁰ + α^{(386 % 255)}x⁹ + α^{(490 % 255)}x⁸ + α^{(417 % 255)}x⁷ + α²⁴³x⁶ + α^{(336 % 255)}x⁵ + α^{(334 % 255)}x⁴ + α^{(391 % 255)}x³

The result is:

α²³⁸x²¹ + α¹⁹⁸x²⁰ + α²¹⁷x¹⁹ + α¹⁴¹x¹⁸ + α⁷⁷x¹⁷ + α¹⁶⁷x¹⁶ + α⁸⁰x¹⁵ + α¹⁰¹x¹⁴ + α¹⁵³x¹³ + α⁶²x¹² + α¹⁷⁰x¹¹ + α¹³⁵x¹⁰ + α¹³¹x⁹ + α²³⁵x⁸ + α¹⁶²x⁷ + α²⁴³x⁶ + α⁸¹x⁵ + α⁷⁹x⁴ + α¹³⁶x³

Now, convert this to integer notation:

11x²¹ + 7x²⁰ + 155x¹⁹ + 21x¹⁸ + 60x¹⁷ + 126x¹⁶ + 253x¹⁵ + 34x¹⁴ + 146x¹³ + 222x¹² + 215x¹¹ + 169x¹⁰ + 92x⁹ + 235x⁸ + 191x⁷ + 125x⁶ + 231x⁵ + 240x⁴ + 79x³

Step 12b: XOR the result with the result from step 11b

Use the result from step 11b to perform the next XOR.

(11 ⊕ 11)x²¹ + (42 ⊕ 7)x²⁰ + (115 ⊕ 155)x¹⁹ + (64 ⊕ 21)x¹⁸ + (120 ⊕ 60)x¹⁷ + (172 ⊕ 126)x¹⁶ + (109 ⊕ 253)x¹⁵ + (210 ⊕ 34)x¹⁴ + (46 ⊕ 146)x¹³ + (181 ⊕ 222)x¹² + (173 ⊕ 215)x¹¹ + (177 ⊕ 169)x¹⁰ + (1 ⊕ 92)x⁹ + (163 ⊕ 235)x⁸ + (139 ⊕ 191)x⁷ + (116 ⊕ 125)x⁶ + (119 ⊕ 231)x⁵ + (242 ⊕ 240)x⁴ + (0 ⊕ 79)x³

The result is:

0x²¹ + 45x²⁰ + 232x¹⁹ + 85x¹⁸ + 68x¹⁷ + 210x¹⁶ + 144x¹⁵ + 240x¹⁴ + 188x¹³ + 107x¹² + 122x¹¹ + 24x¹⁰ + 93x⁹ + 72x⁸ + 52x⁷ + 9x⁶ + 144x⁵ + 2x⁴ + 79x³

Discard the lead 0 term to get:

45x²⁰ + 232x¹⁹ + 85x¹⁸ + 68x¹⁷ + 210x¹⁶ + 144x¹⁵ + 240x¹⁴ + 188x¹³ + 107x¹² + 122x¹¹ + 24x¹⁰ + 93x⁹ + 72x⁸ + 52x⁷ + 9x⁶ + 144x⁵ + 2x⁴ + 79x³

Step 13a: Multiply the Generator Polynomial by the Lead Term of the XOR result from the previous step

Next, multiply the generator polynomial by the lead term of the XOR result from the previous step. The lead term in this case is 45x²⁰. Convert 45x²⁰ to alpha notation. According to the log antilog table, for the integer value 45, the alpha exponent is 18. Therefore 45 = α¹⁸. Multiply the generator polynomial by α¹⁸:

(α¹⁸ * α⁰)x²⁰ + (α¹⁸ * α²¹⁵)x¹⁹ + (α¹⁸ * α²³⁴)x¹⁸ + (α¹⁸ * α¹⁵⁸)x¹⁷ + (α¹⁸ * α⁹⁴)x¹⁶ + (α¹⁸ * α¹⁸⁴)x¹⁵ + (α¹⁸ * α⁹⁷)x¹⁴ + (α¹⁸ * α¹¹⁸)x¹³ + (α¹⁸ * α¹⁷⁰)x¹² + (α¹⁸ * α⁷⁹)x¹¹ + (α¹⁸ * α¹⁸⁷)x¹⁰ + (α¹⁸ * α¹⁵²)x⁹ + (α¹⁸ * α¹⁴⁸)x⁸ + (α¹⁸ * α²⁵²)x⁷ + (α¹⁸ * α¹⁷⁹)x⁶ + (α¹⁸ * α⁵)x⁵ + (α¹⁸ * α⁹⁸)x⁴ + (α¹⁸ * α⁹⁶)x³ + (α¹⁸ * α¹⁵³)x²

The exponents of the alphas are added together. In this case, at least one of the exponents is larger than 255, so perform modulo 255 as follows:

α¹⁸x²⁰ + α²³³x¹⁹ + α²⁵²x¹⁸ + α¹⁷⁶x¹⁷ + α¹¹²x¹⁶ + α²⁰²x¹⁵ + α¹¹⁵x¹⁴ + α¹³⁶x¹³ + α¹⁸⁸x¹² + α⁹⁷x¹¹ + α²⁰⁵x¹⁰ + α¹⁷⁰x⁹ + α¹⁶⁶x⁸ + α^{(270 % 255)}x⁷ + α¹⁹⁷x⁶ + α²³x⁵ + α¹¹⁶x⁴ + α¹¹⁴x³ + α¹⁷¹x²

The result is:

α¹⁸x²⁰ + α²³³x¹⁹ + α²⁵²x¹⁸ + α¹⁷⁶x¹⁷ + α¹¹²x¹⁶ + α²⁰²x¹⁵ + α¹¹⁵x¹⁴ + α¹³⁶x¹³ + α¹⁸⁸x¹² + α⁹⁷x¹¹ + α²⁰⁵x¹⁰ + α¹⁷⁰x⁹ + α¹⁶⁶x⁸ + α¹⁵x⁷ + α¹⁹⁷x⁶ + α²³x⁵ + α¹¹⁶x⁴ + α¹¹⁴x³ + α¹⁷¹x²

Now, convert this to integer notation:

45x²⁰ + 243x¹⁹ + 173x¹⁸ + 227x¹⁷ + 129x¹⁶ + 112x¹⁵ + 124x¹⁴ + 79x¹³ + 165x¹² + 175x¹¹ + 167x¹⁰ + 215x⁹ + 63x⁸ + 38x⁷ + 141x⁶ + 201x⁵ + 248x⁴ + 62x³ + 179x²

Step 13b: XOR the result with the result from step 12b

Use the result from step 12b to perform the next XOR.

(45 ⊕ 45)x²⁰ + (232 ⊕ 243)x¹⁹ + (85 ⊕ 173)x¹⁸ + (68 ⊕ 227)x¹⁷ + (210 ⊕ 129)x¹⁶ + (144 ⊕ 112)x¹⁵ + (240 ⊕ 124)x¹⁴ + (188 ⊕ 79)x¹³ + (107 ⊕ 165)x¹² + (122 ⊕ 175)x¹¹ + (24 ⊕ 167)x¹⁰ + (93 ⊕ 215)x⁹ + (72 ⊕ 63)x⁸ + (52 ⊕ 38)x⁷ + (9 ⊕ 141)x⁶ + (144 ⊕ 201)x⁵ + (2 ⊕ 248)x⁴ + (79 ⊕ 62)x³ + (0 ⊕ 179)x²

The result is:

0x²⁰ + 27x¹⁹ + 248x¹⁸ + 167x¹⁷ + 83x¹⁶ + 224x¹⁵ + 140x¹⁴ + 243x¹³ + 206x¹² + 213x¹¹ + 191x¹⁰ + 138x⁹ + 119x⁸ + 18x⁷ + 132x⁶ + 89x⁵ + 250x⁴ + 113x³ + 179x²

Discard the lead 0 term to get:

27x¹⁹ + 248x¹⁸ + 167x¹⁷ + 83x¹⁶ + 224x¹⁵ + 140x¹⁴ + 243x¹³ + 206x¹² + 213x¹¹ + 191x¹⁰ + 138x⁹ + 119x⁸ + 18x⁷ + 132x⁶ + 89x⁵ + 250x⁴ + 113x³ + 179x²

Step 14a: Multiply the Generator Polynomial by the Lead Term of the XOR result from the previous step

Next, multiply the generator polynomial by the lead term of the XOR result from the previous step. The lead term in this case is 27x¹⁹. Convert 27x¹⁹ to alpha notation. According to the log antilog table, for the integer value 27, the alpha exponent is 248. Therefore 27 = α²⁴⁸. Multiply the generator polynomial by α²⁴⁸:

(α²⁴⁸ * α⁰)x¹⁹ + (α²⁴⁸ * α²¹⁵)x¹⁸ + (α²⁴⁸ * α²³⁴)x¹⁷ + (α²⁴⁸ * α¹⁵⁸)x¹⁶ + (α²⁴⁸ * α⁹⁴)x¹⁵ + (α²⁴⁸ * α¹⁸⁴)x¹⁴ + (α²⁴⁸ * α⁹⁷)x¹³ + (α²⁴⁸ * α¹¹⁸)x¹² + (α²⁴⁸ * α¹⁷⁰)x¹¹ + (α²⁴⁸ * α⁷⁹)x¹⁰ + (α²⁴⁸ * α¹⁸⁷)x⁹ + (α²⁴⁸ * α¹⁵²)x⁸ + (α²⁴⁸ * α¹⁴⁸)x⁷ + (α²⁴⁸ * α²⁵²)x⁶ + (α²⁴⁸ * α¹⁷⁹)x⁵ + (α²⁴⁸ * α⁵)x⁴ + (α²⁴⁸ * α⁹⁸)x³ + (α²⁴⁸ * α⁹⁶)x² + (α²⁴⁸ * α¹⁵³)x¹

The exponents of the alphas are added together. In this case, at least one of the exponents is larger than 255, so perform modulo 255 as follows:

α²⁴⁸x¹⁹ + α^{(463 % 255)}x¹⁸ + α^{(482 % 255)}x¹⁷ + α^{(406 % 255)}x¹⁶ + α^{(342 % 255)}x¹⁵ + α^{(432 % 255)}x¹⁴ + α^{(345 % 255)}x¹³ + α^{(366 % 255)}x¹² + α^{(418 % 255)}x¹¹ + α^{(327 % 255)}x¹⁰ + α^{(435 % 255)}x⁹ + α^{(400 % 255)}x⁸ + α^{(396 % 255)}x⁷ + α^{(500 % 255)}x⁶ + α^{(427 % 255)}x⁵ + α²⁵³x⁴ + α^{(346 % 255)}x³ + α^{(344 % 255)}x² + α^{(401 % 255)}x¹

The result is:

α²⁴⁸x¹⁹ + α²⁰⁸x¹⁸ + α²²⁷x¹⁷ + α¹⁵¹x¹⁶ + α⁸⁷x¹⁵ + α¹⁷⁷x¹⁴ + α⁹⁰x¹³ + α¹¹¹x¹² + α¹⁶³x¹¹ + α⁷²x¹⁰ + α¹⁸⁰x⁹ + α¹⁴⁵x⁸ + α¹⁴¹x⁷ + α²⁴⁵x⁶ + α¹⁷²x⁵ + α²⁵³x⁴ + α⁹¹x³ + α⁸⁹x² + α¹⁴⁶x¹

Now, convert this to integer notation:

27x¹⁹ + 81x¹⁸ + 144x¹⁷ + 170x¹⁶ + 127x¹⁵ + 219x¹⁴ + 223x¹³ + 206x¹² + 99x¹¹ + 101x¹⁰ + 150x⁹ + 77x⁸ + 21x⁷ + 233x⁶ + 123x⁵ + 71x⁴ + 163x³ + 225x² + 154x¹

Step 14b: XOR the result with the result from step 13b

Use the result from step 13b to perform the next XOR.

(27 ⊕ 27)x¹⁹ + (248 ⊕ 81)x¹⁸ + (167 ⊕ 144)x¹⁷ + (83 ⊕ 170)x¹⁶ + (224 ⊕ 127)x¹⁵ + (140 ⊕ 219)x¹⁴ + (243 ⊕ 223)x¹³ + (206 ⊕ 206)x¹² + (213 ⊕ 99)x¹¹ + (191 ⊕ 101)x¹⁰ + (138 ⊕ 150)x⁹ + (119 ⊕ 77)x⁸ + (18 ⊕ 21)x⁷ + (132 ⊕ 233)x⁶ + (89 ⊕ 123)x⁵ + (250 ⊕ 71)x⁴ + (113 ⊕ 163)x³ + (179 ⊕ 225)x² + (0 ⊕ 154)x¹

The result is:

0x¹⁹ + 169x¹⁸ + 55x¹⁷ + 249x¹⁶ + 159x¹⁵ + 87x¹⁴ + 44x¹³ + 0x¹² + 182x¹¹ + 218x¹⁰ + 28x⁹ + 58x⁸ + 7x⁷ + 109x⁶ + 34x⁵ + 189x⁴ + 210x³ + 82x² + 154x¹

Discard the lead 0 term to get:

169x¹⁸ + 55x¹⁷ + 249x¹⁶ + 159x¹⁵ + 87x¹⁴ + 44x¹³ + 0x¹² + 182x¹¹ + 218x¹⁰ + 28x⁹ + 58x⁸ + 7x⁷ + 109x⁶ + 34x⁵ + 189x⁴ + 210x³ + 82x² + 154x¹

Step 15a: Multiply the Generator Polynomial by the Lead Term of the XOR result from the previous step

Next, multiply the generator polynomial by the lead term of the XOR result from the previous step. The lead term in this case is 169x¹⁸. Convert 169x¹⁸ to alpha notation. According to the log antilog table, for the integer value 169, the alpha exponent is 135. Therefore 169 = α¹³⁵. Multiply the generator polynomial by α¹³⁵:

(α¹³⁵ * α⁰)x¹⁸ + (α¹³⁵ * α²¹⁵)x¹⁷ + (α¹³⁵ * α²³⁴)x¹⁶ + (α¹³⁵ * α¹⁵⁸)x¹⁵ + (α¹³⁵ * α⁹⁴)x¹⁴ + (α¹³⁵ * α¹⁸⁴)x¹³ + (α¹³⁵ * α⁹⁷)x¹² + (α¹³⁵ * α¹¹⁸)x¹¹ + (α¹³⁵ * α¹⁷⁰)x¹⁰ + (α¹³⁵ * α⁷⁹)x⁹ + (α¹³⁵ * α¹⁸⁷)x⁸ + (α¹³⁵ * α¹⁵²)x⁷ + (α¹³⁵ * α¹⁴⁸)x⁶ + (α¹³⁵ * α²⁵²)x⁵ + (α¹³⁵ * α¹⁷⁹)x⁴ + (α¹³⁵ * α⁵)x³ + (α¹³⁵ * α⁹⁸)x² + (α¹³⁵ * α⁹⁶)x¹ + (α¹³⁵ * α¹⁵³)x⁰

The exponents of the alphas are added together. In this case, at least one of the exponents is larger than 255, so perform modulo 255 as follows:

α¹³⁵x¹⁸ + α^{(350 % 255)}x¹⁷ + α^{(369 % 255)}x¹⁶ + α^{(293 % 255)}x¹⁵ + α²²⁹x¹⁴ + α^{(319 % 255)}x¹³ + α²³²x¹² + α²⁵³x¹¹ + α^{(305 % 255)}x¹⁰ + α²¹⁴x⁹ + α^{(322 % 255)}x⁸ + α^{(287 % 255)}x⁷ + α^{(283 % 255)}x⁶ + α^{(387 % 255)}x⁵ + α^{(314 % 255)}x⁴ + α¹⁴⁰x³ + α²³³x² + α²³¹x¹ + α^{(288 % 255)}x⁰

The result is:

α¹³⁵x¹⁸ + α⁹⁵x¹⁷ + α¹¹⁴x¹⁶ + α³⁸x¹⁵ + α²²⁹x¹⁴ + α⁶⁴x¹³ + α²³²x¹² + α²⁵³x¹¹ + α⁵⁰x¹⁰ + α²¹⁴x⁹ + α⁶⁷x⁸ + α³²x⁷ + α²⁸x⁶ + α¹³²x⁵ + α⁵⁹x⁴ + α¹⁴⁰x³ + α²³³x² + α²³¹x¹ + α³³x⁰

Now, convert this to integer notation:

169x¹⁸ + 226x¹⁷ + 62x¹⁶ + 148x¹⁵ + 122x¹⁴ + 95x¹³ + 247x¹² + 71x¹¹ + 5x¹⁰ + 249x⁹ + 194x⁸ + 157x⁷ + 24x⁶ + 184x⁵ + 210x⁴ + 132x³ + 243x² + 245x¹ + 39

Step 15b: XOR the result with the result from step 14b

Use the result from step 14b to perform the next XOR.

(169 ⊕ 169)x¹⁸ + (55 ⊕ 226)x¹⁷ + (249 ⊕ 62)x¹⁶ + (159 ⊕ 148)x¹⁵ + (87 ⊕ 122)x¹⁴ + (44 ⊕ 95)x¹³ + (0 ⊕ 247)x¹² + (182 ⊕ 71)x¹¹ + (218 ⊕ 5)x¹⁰ + (28 ⊕ 249)x⁹ + (58 ⊕ 194)x⁸ + (7 ⊕ 157)x⁷ + (109 ⊕ 24)x⁶ + (34 ⊕ 184)x⁵ + (189 ⊕ 210)x⁴ + (210 ⊕ 132)x³ + (82 ⊕ 243)x² + (154 ⊕ 245)x¹ + (0 ⊕ 39)x⁰

The result is:

0x¹⁸ + 213x¹⁷ + 199x¹⁶ + 11x¹⁵ + 45x¹⁴ + 115x¹³ + 247x¹² + 241x¹¹ + 223x¹⁰ + 229x⁹ + 248x⁸ + 154x⁷ + 117x⁶ + 154x⁵ + 111x⁴ + 86x³ + 161x² + 111x¹ + 39

Discard the lead 0 term to get:

213x¹⁷ + 199x¹⁶ + 11x¹⁵ + 45x¹⁴ + 115x¹³ + 247x¹² + 241x¹¹ + 223x¹⁰ + 229x⁹ + 248x⁸ + 154x⁷ + 117x⁶ + 154x⁵ + 111x⁴ + 86x³ + 161x² + 111x¹ + 39

Use the terms of the remainder as the error correction codewords

The division has been performed 15 times, which is the number of terms in the message polynomial. This means that the division is complete and the terms of the above polynomial are the error correction codewords to use for the original message polynomial:

213  199  11  45  115  247  241  223  229  248  154  117  154  111  86  161  111  39