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 \(\eta = L_{min}/L_{avg}\), where \(L_{min}\) is related to entropy \(H(S)\) (e.g., \(L_{min} \ge H(S)\) for uniquely decodable codes). Average length is \(L_{avg}\). Higher efficiency means \(L_{avg}\) is closer to \(L_{min}\). So 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 \(t\) errors where \(2t+1 \le 4 \Rightarrow 2t \le 3 \Rightarrow t \le 1.5\). So it can correct single errors (\(t=1\)). It cannot correct double errors. FALSE. (For error detection, \(d_{min} \ge e+1\), where \(e\) is number of errors detected. \(d_{min}=4\) can detect \(e=3\) errors). (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., \(g(0) \neq 0\)). A polynomial \(g(x)\) is irreducible if it cannot be factored into polynomials of lower degree over the given field. The relationship between irreducible and separable for polynomials depends on the field characteristics. In fields of characteristic 0 or for \(g'(x) \neq 0\), irreducible implies separable. For finite fields of characteristic p, a polynomial is separable if and only if its formal derivative is non-zero. An irreducible polynomial \(p(x)\) over a perfect field is separable. For binary codes (field GF(2)), an irreducible polynomial \(g(x)\) is separable if its derivative \(g'(x) \neq 0\). Every irreducible polynomial over GF(2) other than \(x\) is separable. If \(g(x)=x\), it's irreducible but not separable (as \(g(0)=0\)). Assuming "code is irreducible" refers to its generator polynomial \(g(x)\) being irreducible and not simply \(x\). Then, typically, irreducible implies separable in contexts relevant to coding theory (e.g., over finite fields used in practice). This statement is more nuanced from abstract algebra but is often taken as true in specific coding contexts. Compared to (a), (b), (c) which are definitively false based on common definitions, (d) is the most likely "true" statement. Given the options, (a), (b), and (c) are clearly false under standard definitions. Therefore, (d) is assumed to be the true statement, possibly referring to a specific context in coding theory where this holds. \[ \boxed{\text{When a code is irreducible, it is also separable}} \]