Computer Networks · TEE · practice

Block coding, Hamming distance and CRC

Extra mathematical models

TEE · Complete six-unit quick reference · current syllabus 702AI0C023

r=n-k,\quad R_c=k/nLaTeX: r=n-k,\quad R_c=k/n

k data bits, n codeword bits, r redundant bits; coding rate is dimensionless.

d(a,b)=\operatorname{weight}(a\oplus b)LaTeX: d(a,b)=\operatorname{weight}(a\oplus b)

Count differing bit positions in equal-length words.

s=d_{\min}-1,\quad t=\lfloor(d_{\min}-1)/2\rfloorLaTeX: s=d_{\min}-1,\quad t=\lfloor(d_{\min}-1)/2\rfloor

Separate maximum guaranteed detection and correction counts; all distinct pairs define dmin.

2^r\ge m+r+1LaTeX: 2^r\ge m+r+1

Smallest r for m data bits in single-error Hamming construction; parity positions 1,2,4,….

R(x)=x^rM(x)\bmod G(x),\quad T=x^rM\oplus RLaTeX: R(x)=x^rM(x)\bmod G(x),\quad T=x^rM\oplus R

Generator degree r; append r zeros and use modulo-2 XOR division. Receiver divides T directly.

CRC and receiver-check working