This note corrects an error in the paper, "An Efficient Game Form for Multi-Rate Multicast Service Provisioning" (ibid., vol. 30, no. 11,pp. 2093-2104, December 2012) which presents a rate allocation mechanism for multi-rate multicast service provisioning in networks with arbitrary topology and strategic users.

We consider the decentralized bandwidth/rate allocation problem in multi-rate multicast service provisioning with strategic users. We demonstrate that such a situation is the combination of a market problem and a public goods problem. We present a mechanism/game form which possesses the following properties when the users' utilities are concave: (1) It implements in Nash equilibria the solution of the corresponding centralized rate allocation problem in multi-rate multicast service provisioning. (2) It is individually rational. (3) It is budget-balanced at all Nash equilibria of the game induced by the mechanism/game form as well as at all off equilibrium messages/strategies that result in feasible allocations.

In this paper, we study the redundancy of Huffman codes. In particular, we consider sources for which the probability of one of the source symbols is known. We prove a conjecture of Ye and Yeung regarding the upper bound on the redundancy of such Huffman codes, which yields in a tight upper bound. We also derive a tight lower bound for the redundancy under the same assumption. We further apply the method introduced in this paper to other related problems. It is shown that several other previously known bounds with different constraints follow immediately from our results.

In this study, the authors consider the class of anti-uniform Huffman (AUH) codes. The authors derived tight lower and upper bounds on the average codeword length, entropy and redundancy of finite and infinite AUH codes in terms of the alphabet size of the source. These bounds are tighter than similar bounds. Also a tight upper bound on the entropy of AUH codes is presented in terms of the average cost of the code. The Fibonacci distribution is introduced, which plays a fundamental role in AUH codes. It is shown that such distributions maximise the average length and the entropy of the code for a given alphabet size. The authors also show that the minimum average cost of a code is achieved by an AUH codes in a highly unbalanced cost regime.

A source ={s1, s2, …} having a binary Huffman code with code-word lengths satisfying l1=1, l2=2, … is called an anti-uniform source. If l1=1, l2=2, … , li=i, then the source is called an i-level partially anti-uniform source. The redundancy, expected codeword length and entropy of anti-uniform sources are dealt here. A tight upper bound is derived for the expected codeword length L of anti-uniform sources. It is shown that L does not exceed $\surd{5}+3/2$. For each $1\lt L\le \lpar \surd{5}+3\rpar /2$, an anti-uniform distribution achieving maximum entropy H(P)max=L log L−(L−1)log(L−1) is introduced. This shows that the maximum entropy achieved by anti-uniform distributions does not exceed 2.512. It is shown that the range of redundancy values for i-level partially anti-uniform sources with distribution {pi} is an interval of length ∑j=i+1pj. This results in a realistic approximation for the redundancy of these sources.

A source S = {s1, s2, ...} having a binary Huffman code with codeword lengths satisfying l1= 1, l2= 2, ...(and ln-1= ln = n - 1 when |S| = n) is called an antiuniform source. If l1= 1, l2= 2, ..., li = i, then the source is called an i-level partially antiuniform source. In this paper the redundancy, the expected codeword lengths, and the entropy of partially antiuniform sources is studied. It is shown that the range of the redundancy R of a given i-level partially antiuniform source with distribution {pi} is an interval of length pi+1 + ... +pn. This results in a realistic approximation for R. An upper bound is derived for the expected codeword lengths L of antiuniform sources. It is shown that L is less than three. The entropy H of any antiuniform source with average codeword lengths L is bounded above by LlogL - (L - 1) log(L - 1) and hence it does not exceed 2.76.

In this paper we study the redundancy of Huffman codes. In particular, we consider sources for which the probability of one of the source symbols is known. We prove a conjecture of Ye and Yeung regarding the upper bound on the redundancy of such Huffman codes, which yields in a tight upper bound. We also derive a tight lower bound for the redundancy under the same assumption. We further apply the method introduced in this paper to other related problems. It is shown that several other previously known bounds with different constraints follow immediately from our results.