Question

Which one of the following is true?

• A. The efficiency of Huffman code is linearly proportional to average length of code
• B. Huffman code is also known as maximum redundancy code
• C. A code with Hamming distance 4 is capable of double error correction
• D. When a code is irreducible, it is also separable

Hint:

Huffman coding minimizes average codeword length, hence minimizes redundancy.
Error correction capability: \(d_{min} \ge 2t+1\) for \(t\)-error correction.
Error detection capability: \(d_{min} \ge e+1\) for \(e\)-error detection.
"Irreducible" and "separable" are properties of polynomials, relevant to algebraic codes like cyclic codes.

Answer 1

The Correct Option is D

Let's analyze each statement: 

• (a) "The efficiency of Huffman code is linearly proportional to average length of code":

Efficiency \( \eta = \frac{L_{min}}{L_{avg}} \), where \(L_{min}\) is related to entropy \(H(S)\) (e.g., \(L_{min} \ge H(S)\) for uniquely decodable codes), and \(L_{avg}\) is the average length of the code. Higher efficiency means \(L_{avg}\) is closer to \(L_{min}\). Therefore, efficiency is inversely related to \(L_{avg}\) for a fixed source, not linearly proportional. FALSE.

• (b) "Huffman code is also known as maximum redundancy code":

Huffman coding is an algorithm for constructing optimal prefix codes, meaning it aims to minimize the average codeword length, and thus minimize redundancy (\(Redundancy = L_{avg} - H(S)\)). So, it's a minimum redundancy code, not maximum. FALSE.

• (c) "A code with Hamming distance 4 is capable of double error correction":

For error correction, the minimum Hamming distance \(d_{min}\) must satisfy \(d_{min} \ge 2t + 1\), where \(t\) is the number of errors that can be corrected. If \(t = 2\) (double error correction), then \(d_{min} \ge 2(2) + 1 = 5\). A code with Hamming distance 4 can correct only single errors, not double errors. FALSE.

• (d) "When a code is irreducible, it is also separable":

This statement refers to properties of polynomial codes (like cyclic codes). A polynomial code is defined by its generator polynomial \(g(x)\). A code is separable if \(g(x)\) does not have \(x\) as a factor (i.e., \(g(0) \neq 0\)).

An irreducible polynomial \(g(x)\) cannot be factored into polynomials of lower degree over the given field. The relationship between irreducible and separable depends on the field characteristics. Over finite fields, irreducibility typically implies separability. In coding theory, irreducible polynomials over fields like GF(2) are typically separable. TRUE in the context of binary codes.

Given the options, (a), (b), and (c) are clearly false under standard definitions. Therefore, (d) is the true statement, possibly referring to a specific context in coding theory where this holds.

Final Answer:

When a code is irreducible, it is also separable