Takes 4 information bits and generates 7 coded bits (called a codeword). In this 7-bit codeword representation [x, x, ...x], x represents the Most Significant Bit (MSB) and x represents the Least Significant Bit (LSB). The first four MSBs [x, x, x, x4] in each of these codewords represent the 4-bit information sequence. The remaining three bits [x, x, x] represent the parity check-bits.
a. Find the parity matrix H and generate a syndrome table.
b. Let the received codeword be [1 1 1 1 0 1 1]. Assume that this received codeword has one bit in error. What is the receiver's interpretation of the 4 information bits that were transmitted?
c. Assume that the transmitter used the 4 information bits that were derived in part (b) above. What is the receiver's interpretation of the 4 information bits that were transmitted if the received codeword is [1 0 1 1 0 1 1]?
d. Suppose the receiver does not have adequate memory to store both the H matrix and the syndrome table, but only has sufficient memory to store the H matrix. Can the receiver locate the error in item (b) above without the syndrome table and without additional matrix multiplications than what is necessary for item (b)?
1000111
0100101
G = 0010011
0001110