A clean codeword divides evenly by the generator. The verify matrix recomputes that zero remainder from the transmitted bits.

highlighted = computed this step

Why verification divides the whole codeword

A receiver divides the complete codeword, not just the original message. A clean codeword is divisible by the generator.

remainder=000\text{remainder}=000

Remainder zero means no detected error

Dividing codeword 11010011101100100 by 1011 leaves remainder 000. That means no error is detected.

11010011101100100mod1011=00011010011101100100\bmod1011=000
Verify codewordGF(2) division rows are recomputed from the exact bits.Verify codewordverify by 1011dividend1101001110110010011010011101100100xor 0101101100011101100100xor 1101100111011101100100xor 2101100010111101100100xor 3101100000001101100100skip 400000001101100100skip 500000001101100100skip 600000001101100100xor 7101100000000110100100xor 8101100000000011000100xor 9101100000000001110100xor 10101100000000000101100xor 11101100000000000000000skip 1200000000000000000skip 1300000000000000000remainder000000

Why this works

The sender chose the CRC so message plus CRC is divisible by the generator. Verification checks that divisibility again.

zero remainder=000\text{zero remainder}=000
Verify codewordGF(2) division rows are recomputed from the exact bits.Verify codewordverify by 1011dividend1101001110110010011010011101100100xor 0101101100011101100100xor 1101100111011101100100xor 2101100010111101100100xor 3101100000001101100100skip 400000001101100100skip 500000001101100100skip 600000001101100100xor 7101100000000110100100xor 8101100000000011000100xor 9101100000000001110100xor 10101100000000000101100xor 11101100000000000000000skip 1200000000000000000skip 1300000000000000000remainder000000

Summary

CRC verification recomputes divisibility over the complete codeword and expects a zero remainder. Polynomial arithmetic over the exact bits; timing/throughput is not modeled here.

remainder=000\text{remainder}=000
Verify codewordGF(2) division rows are recomputed from the exact bits.Verify codewordverify by 1011dividend1101001110110010011010011101100100xor 0101101100011101100100xor 1101100111011101100100xor 2101100010111101100100xor 3101100000001101100100skip 400000001101100100skip 500000001101100100skip 600000001101100100xor 7101100000000110100100xor 8101100000000011000100xor 9101100000000001110100xor 10101100000000000101100xor 11101100000000000000000skip 1200000000000000000skip 1300000000000000000remainder000000