We derive expressions for the average distance distributions in several ensembles of regular low-density parity-check codes (LDPC). Among these ensembles are the standard one defined by matrices having given column and row sums, ensembles defined by matrices with given column sums or given row sums, and an ensemble defined by bipartite graphs.
We derive asymptotic expressions for the average distance distributions in ensembles of irregular low-density paritycheck (LDPC) codes. The ensembles are defined by matrices with given profiles of column and row sums. Index Terms-Distance distributions, low-density parity-check (LDPC) codes. I. INTRODUCTION T HE low-density parity-check (LDPC) codes are famous because of their performance in the vicinity of the Shannon limit under iterative decoding of modest complexity. However, this phenomenon is still far from being completely understood. One of the important parameters helping in analysis of codes' performance under maximum-likelihood decoding is their distance distribution. Such estimates are important in analyzing achievable limits of performance of LDPC codes, and optimization of their parameters. The problem of estimation distance distribution of regular codes (i.e., codes defined by parity-check matrices with fixed column and row sums) was addressed in many papers, starting with the original Gallager's work [2], see for most general results [5]. Use of such estimates to bounding performance of LDPC codes was discussed in [2], [7], [9], [13]. Recently, it was found that irregular LDPC codes (i.e., codes defined by parity-check matrices with several possible values of column and row sums) perform better than the regular codes under iterative decoding [6], [11], [12], [14]. In this paper, we solve the problem of estimation of the average distance distribution in ensembles of irregular LDPC codes. This question was addressed independently in [4], where an implicit expression for such distributions was given. Here we give an explicit formula describing the average distance distributions. The paper is organized as follows. We start by defining the considered ensemble of irregular codes, and define the average distance distribution in this ensemble. Then we count the number of matrices belonging to the ensemble of special shape, yielding that the defined codes contain the word having ones on the first positions and zeros on the remaining positions.
We call n a near-perfect number if n is the sum of all of its proper divisors, except for one of them, which we term the redundant divisor. For example, the representationshows that 12 is near-perfect with redundant divisor 4. Nearperfect numbers are thus a very special class of pseudoperfect numbers, as defined by Sierpiński. We discuss some rules for generating near-perfect numbers similar to Euclid's rule for constructing even perfect numbers, and we obtain an upper bound of x 5/6+o(1) for the number of near-perfect numbers in [1, x], as x → ∞.
In a recent post on the Seqfan list the third author proposed a conjecture concerning the summatory function of odious numbers (i.e., of numbers whose sum of binary digits is odd), and its analog for evil numbers (i.e., of numbers whose sum of binary digits is even). We prove these conjectures here. We will also study the sequences of "generalized" odious and evil numbers, and their iterations, giving in particular a characterization of the sequences of usual odious and evil numbers in terms of functional equations satisfied by their compositions.
We consider the number of permutations with prescribed up-down structure as a function of two arguments: the number n of elements and the introduced up-down index k of a permutation. We consider sets of permutations for which k is a fixed number and when k is a function of n. In the first case the number of permutations is a polynomial in n, the degree of which is defined by k.
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