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:

  • $\displaystyle d(x,y)$

  • $\displaystyle d(\code)$

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

  • $\displaystyle P = [A \ I_r ]$