Error correcting codes are very significant in the transmission of binary information from the source to the recipient. The presence of noise in a communication channel can cause errors in the transmission of binary information. Error correcting codes are used to correct the detected errors in the transmission of binary information. The two main error correcting codes used are the generator matrix and the parity check matrix (Huffman and Vera 57).There are only specific sequences, known as the code words, which are transmitted. Generator matrix is used to generate the code word for transmission while parity check matrix is used to check if the received binary sequence exists in the code word dictionary.
Generator matrix
If C is a code of {n,k}q, then a generator matrix G for code C is any k x n matrix with entries in Fq so that the rows of the generator matrix G forms a basis for C.
That is C= {xG|x EFqk}
The rows in a generator matrix are used to form a basis for a linear code. It is used to generate a code word for a message by multiplying the message matrix with the generator matrix (Huffman and Vera 62).
Also, If C is a {n,k}q code, any collection of k that is linearly independent columns of the code C is known as an information set for C. If G is a generator matrix for C, then an information set for code word C is a collection of integers {i1 to in}such that the conforming columns of the generator matrix G are linearly sovereign vectors.
Example
If C is the code word with the generator matrix
1 000011010010100101100001111
Then C is a 7 by 4 code. Therefore the information sets includes: {1234}{1235}{1345}{1245} {1236} {1237}
Parity check matrix
If C is a code word of the form {n,k}q, then a parity check matrix for the code word C is a matrix of the form (n-k)x n H above Fq such that
C= (cEFqn: HcT=0)
That is to say that a parity check matrix, H of a code word C is regarded as a generator matrix of the dual code, CT . That implies that a code word c can only belong to C under the condition that the matrix-vector product is equal to zero; HcT=0. It is imperative to note that the parity check matrix for a specific code word C can be attained from its generator matrix (Huffman and Vera 67).
Example
If G = 10|10101|110 then the parity check is attained as
H= 11|10001|01010|001
Works Cited
Huffman, W. Cary, and Vera Pless. Fundamentals of error-correcting codes. Cambridge University press, 2010.
