Combinatorics

Summary.

• how to make a code from a design

• Important definitions:

  • binary string

  • code, codeword

  • Hamming distance

  • minimum distance of a code

  • detect errors, correct errors

  • encode, decode

  • generator matrix

  • parity-check matrix

  • binary linear code of type \((n,k)\)

  • Hamming code

• Notation:

  • \(\displaystyle d(x,y)\)

  • \(\displaystyle d(\code)\)

  • \(\displaystyle G = \begin{bmatrix}I_k \\ A \end{bmatrix}\)

  • \(\displaystyle P = [A \ I_r ]\)