Weight - number of non-zero entries in a code

Min Distance = Min Weight

Binary Hamming Codes

A hamming code $Hamming(n,k)$ has $n$ bits, of which $k$ are information bits.

Through maths, there are $n-k$ check bits.

There are as many rows as there are checkbits ($n-k$ rows), and as many columns as there are bits ($n$ columns)

For example, $Hamming(7,4)$

1
2
3

1 0 1 0 1 0 1
0 1 1 0 0 1 1
0 0 0 1 1 1 1
  • $7$ bits
  • $4$ information bits
  • $3$ check bits

So it a $7$ x $3$ matrix

Binary Hamming Error-correction

A code $C$ that

  • is binary with the code alphabet ${0, 1}$
  • has fixed length $n$ codewords $x = x_1 … x_n$
  • is only single-error correcting
  • provides user-friendly error-correcting
  • uses $m$ independent linear parity checks

Read about Parity Check Matrices
Read lecture three