\begin{proof}
We must show that if there exists a uniquely decodable code with codeword lengths ( l_1, l_2, \ldots, l_q ), then the inequality holds.
Suppose ( C ) is a uniquely decodable code with codeword lengths ( l_1, l_2, \ldots, l_q ). Consider a sequence of codewords produced by ( C ) that is long enough to contain any possible concatenation of ( n ) codewords (for some ( n ) large enough). Let's denote the set of all such sequences as ( S_n ).
Because ( C ) is uniquely decodable, each sequence in ( S_n ) corresponds to a unique sequence of source symbols. The number of different sequences in ( S_n ) is at least ( 2^{nl} ), where ( l ) is the average length of the codewords, not the smallest length. This correction accounts for the diversity of codeword lengths and their impact on the encoding process. The original statement about the smallest length ( l ) and its role in generating sequences of ( nl ) may lead to confusion, as it implies a restriction to sequences of the shortest codeword, which is not accurate in the context of uniquely decodable codes.
The total number of bits required to represent all sequences in ( S_n ) can indeed be calculated as ( (2^{-l_1} + 2^{-l_2} + \ldots + 2^{-l_q})^n ), considering the combinatorial possibilities of all codeword concatenations.
Given the unique decodability, the representation of sequences in a binary tree is a helpful visual. However, the conclusion that the number of sequences in ( Sn ) cannot exceed ( 2^{nl{max}} ) needs clarification. The argument should emphasize that the binary representation's expansiveness—dictated by the longest codeword—serves as an upper bound for encoding distinct sequences, thereby ensuring that ( S_n )'s diversity does not overflow the binary system's capacity.
The leap to ( 2^{l_{max}} ) being always less than or equal to 1 is incorrect and misleading. The correction should highlight that the comparison made in the proof actually aims to establish a relationship between the sum of reciprocals and the code's capacity to accommodate uniquely decodable sequences without exceeding the binary representation limit, leading to the corrected conclusion:
This conclusion directly follows from the premise that the sum of the probabilities (represented by ( 2^{-l_i} )) of all possible codeword sequences should not exceed the total probability space (1), ensuring unique decodability.
To prove sufficiency, the argument that a prefix-free code construction, which satisfies the inequality, ensures unique decodability is valid and well-stated. This section accurately reflects the logical flow from the theoretical foundation provided by McMillan's Inequality to practical code construction, affirming that prefix-free codes represent a special case of uniquely decodable codes, thereby fulfilling the inequality's conditions.
\end{proof}
This revised explanation aims to correct misunderstandings and provide a clearer, more accurate depiction of McMillan's Inequality and its proof, reinforcing its critical role in understanding the structure and capabilities of uniquely decodable codes.
\begin{proof} We must show that if there exists a uniquely decodable code with codeword lengths ( l_1, l_2, \ldots, l_q ), then the inequality holds.
Suppose ( C ) is a uniquely decodable code with codeword lengths ( l_1, l_2, \ldots, l_q ). Consider a sequence of codewords produced by ( C ) that is long enough to contain any possible concatenation of ( n ) codewords (for some ( n ) large enough). Let's denote the set of all such sequences as ( S_n ).
Because ( C ) is uniquely decodable, each sequence in ( S_n ) corresponds to a unique sequence of source symbols. The number of different sequences in ( S_n ) is at least ( 2^{nl} ), where ( l ) is the average length of the codewords, not the smallest length. This correction accounts for the diversity of codeword lengths and their impact on the encoding process. The original statement about the smallest length ( l ) and its role in generating sequences of ( nl ) may lead to confusion, as it implies a restriction to sequences of the shortest codeword, which is not accurate in the context of uniquely decodable codes.
The total number of bits required to represent all sequences in ( S_n ) can indeed be calculated as ( (2^{-l_1} + 2^{-l_2} + \ldots + 2^{-l_q})^n ), considering the combinatorial possibilities of all codeword concatenations.
Given the unique decodability, the representation of sequences in a binary tree is a helpful visual. However, the conclusion that the number of sequences in ( Sn ) cannot exceed ( 2^{nl{max}} ) needs clarification. The argument should emphasize that the binary representation's expansiveness—dictated by the longest codeword—serves as an upper bound for encoding distinct sequences, thereby ensuring that ( S_n )'s diversity does not overflow the binary system's capacity.
The leap to ( 2^{l_{max}} ) being always less than or equal to 1 is incorrect and misleading. The correction should highlight that the comparison made in the proof actually aims to establish a relationship between the sum of reciprocals and the code's capacity to accommodate uniquely decodable sequences without exceeding the binary representation limit, leading to the corrected conclusion:
[ 2^{-l_1} + 2^{-l_2} + \ldots + 2^{-l_q} \leq 1 ]
This conclusion directly follows from the premise that the sum of the probabilities (represented by ( 2^{-l_i} )) of all possible codeword sequences should not exceed the total probability space (1), ensuring unique decodability.
To prove sufficiency, the argument that a prefix-free code construction, which satisfies the inequality, ensures unique decodability is valid and well-stated. This section accurately reflects the logical flow from the theoretical foundation provided by McMillan's Inequality to practical code construction, affirming that prefix-free codes represent a special case of uniquely decodable codes, thereby fulfilling the inequality's conditions.
\end{proof}
This revised explanation aims to correct misunderstandings and provide a clearer, more accurate depiction of McMillan's Inequality and its proof, reinforcing its critical role in understanding the structure and capabilities of uniquely decodable codes.