Math 350: Applied Algebra: Codes and Ciphers Spring 2009
Linear Binary Codes and the Hamming Codes
Assumptions for all codes in today’s work:
Our codes will be binary codes.
Our codes will be BLOCK codes – which means that all the
codewords in a particular code have the same length (number of
digits).
Noise or interference may change any digit from a 0 to a 1 or vice
verse – but noise will not cause a codeword to drop or add digits.
Our codes will be LINEAR.
C1 = {00, 01, 10, 11} C2 = {0000, 0101, 1010, 1111}
C3 = { 00000, 11111} C4 = {000, 101, 011, }
Let n represent the length of our block codes
k = the number of those digits that are information digits
Define the INFORMATION RATE of a code:
R= k/n = the proportion of digits that are information
digits.
Maximum Likelihood decoding
The Hamming distance between (c1, c2) =d(c1,c2) =number of
positions in which their digits differ.
The MINIMUM DISTANCE of a code is the smallest distance
between any two distinct codewords of the code.
What are the minimum distances of codes C1, C2, C3, C4?
