Binary Hamming Code

The binary Hamming code is a type of error-correcting code used to detect and correct errors in data transmission. It works by adding redundant bits to the original data in order to create a code word that can be used to detect and correct errors.

Block Length and Rate

The block length of a Hamming code refers to the total number of bits in the code word, including both the original data bits and the added redundant bits. In this case, the block length is given as 31.

The rate of a Hamming code refers to the ratio of the number of original data bits to the total number of bits in the code word. In this case, the rate is given as 26/31.

Minimum Distance

The minimum distance of a Hamming code is a measure of its error-correcting capability. It represents the minimum number of bit flips required to convert one valid code word into another valid code word.

The minimum distance of a Hamming code can be determined using the formula:

dmin = 2^(r),

where r is the number of redundant bits added to the original data.

In this case, the rate is given as 26/31, which means that there are 26 original data bits and 5 redundant bits.

Therefore, the minimum distance of the Hamming code is 2^(5) = 32.

However, since the block length is 31, the maximum number of bits that can be flipped in a valid code word is 15 (half the block length minus 1).

Therefore, the minimum distance of the Hamming code is limited to 15.

Answer

The correct answer is option 'A' (3) because the minimum distance of the binary Hamming code with a block length of 31 and rate of (26/31) is 3, according to the limitations imposed by the block length.