Error Control Coding

Syllabus: Fundamentals of error correction, Hamming codes, CRC.

4 hand-picked visual resources on this topic — under each card, look for See it. Every link is checked against the live site before it ships.

Linear block codes and minimum distance

Everything about detection and correction capability follows from $d_{\min}$ , which for a linear code equals the smallest Hamming weight among its non-zero codewords — so you never have to compare all pairs.

$d_{\min}=\min_{c\neq0}w(c)$
linear codes only
$\text{detect }s\text{ errors: }d_{\min}\ge s+1$
$\text{correct }t\text{ errors: }d_{\min}\ge2t+1\ \Rightarrow\ t=\left\lfloor\dfrac{d_{\min}-1}{2}\right\rfloor$
$C=mG,\qquad S=rH^T$
generator and parity-check matrices; a zero syndrome means no detected error
$GH^T=0,\qquad \text{rate}=k/n$

See it

Hamming codes and CRC

A Hamming code is the single-error-correcting code with the fewest parity bits: $r$ parity bits address $2^r-1$ positions, so $n=2^r-1$

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Error Control Coding

Syllabus: Fundamentals of error correction, Hamming codes, CRC.

4 hand-picked visual resources on this topic — under each card, look for See it. Every link is checked against the live site before it ships.

Linear block codes and minimum distance

$d_{\min}=\min_{c\neq0}w(c)$
linear codes only
$\text{detect }s\text{ errors: }d_{\min}\ge s+1$
$\text{correct }t\text{ errors: }d_{\min}\ge2t+1\ \Rightarrow\ t=\left\lfloor\dfrac{d_{\min}-1}{2}\right\rfloor$
$C=mG,\qquad S=rH^T$
generator and parity-check matrices; a zero syndrome means no detected error
$GH^T=0,\qquad \text{rate}=k/n$

See it

Hamming codes and CRC

A Hamming code is the single-error-correcting code with the fewest parity bits: $r$ parity bits address $2^r-1$ positions, so $n=2^r-1$