Different Examples

u c  
00 00000
01 01101
10 10110
11 11011

In this case, the first ten lines of the 32-row decoding table are as follows

Because the size of the memory is limited, table lookup decoding cannot be used generally. If the value of k is small (k << n) then there are only few code words even in the case of long code lengths. This enables to decode the code word by computing the distances code word by code word.

  Let us take the code used in example 1.  It can be seen that the coding can be done by matrix multiplication c = uG where

    G is a binary matrix of size 2x5. In this simple case the set of codes consists of the all-zero vector, the first and the second row the G matrix, and coordinate - by - coordinate modulo-2 (XOR) sum of the two rows. Thus the elements of the C code are generated as linear combinations of the rows of the matrix G. The code C, as a set of binary vectors, forms a linear space. Generalizing this remark we can understand the term of the linear code.