Final answer:
The condition for a code to correct 'k' or fewer errors is that the minimum distance between any two codewords should be '2k+1' or more.
A single error correcting code with 'm=4' and 'n=7' is the Hamming(7,4) code. The parity bits in the Hamming code are calculated by XORing specific combination of data bits.
Step-by-step explanation:
In coding theory, a code is said to correct 'k' or fewer errors if it can detect and correct up to 'k' errors in the received message.
The condition for a code to correct 'k' or fewer errors is that the minimum distance between any two codewords should be '2k+1' or more. This means that the difference in bit positions between any two codewords should be at least '2k+1'.
For a single error correcting code with 'm=4' and 'n=7', we can use a Hamming code. The Hamming(7,4) code is a well-known single error correcting code. It has 4 data bits ('m=4') and 7 total bits ('n=7').
The parity bits in the Hamming code are calculated by XORing specific combination of data bits. The encoded bit string is obtained by appending the parity bits to the data bits.
For example, '1010' can be encoded as '0111010' where the first four bits are the data bits and the last three bits are the parity bits.