A reduced formula for the precise number of (0,1)-matrices in A(R,S)

Improved decoding and error floor analysis of staircase codes

“…The percentage of solvable stall patterns was obtained in simulation by applying the bit-flip operation as described in Algorithm 1 on a dedicated channel designed to insert stall patterns. P C,old and P C,new give the approximation of the contribution to the error floor of (9) and (10) respectively. P f loor,new gives the contribution considering the percentage of stall patterns that can be solved for the given parameters and the error floor is the sum of this column.…”

“…The same arguments as above hold for the limits of the sum. When the last column is reached , i.e., the case (18), the function A ([w 1 , ..., w K ], [w K+1 , ..., w K+L ]) from [12] (see also [10]) gives the number of matrices corresponding to the unique weight vector w of size K + L. Note that while this notation offers a mathematically correct way to calculate the cardinality, when implemented it would be beneficial to call the function A (r, b) as few times as possible, as it is the computationally most complex part. As permutations of the input vectors lead to the same result, many calls of this function can be avoided by, e.g., maintaining a look-up table.…”

“…The problem of finding the number of stall patterns of weight ε within K rows and L columns is equivalent to the problem of finding the number of binary K × L matrices of weight ε and a given weight in each row and column. A solution to this combinatorial problem is given in [12] (see also [10]). The function denoted by A (r, s) takes a vector r = [r 1 , r 2 , ..., r K ] containing the row weights and s = [s 1 , s 2 , ..., s L ] containing the column weights and returns the number of distinct binary matrices that meet the weight restrictions on the rows and columns.…”

Exact sampling and counting for fixed-margin matrices

“…In addition, several other algorithms have been presented for finding N (p, q) [such as Johnsen and Straume (1987), Wang (1988), Wang and Zhang (1998), Pérez-Salvador et al (2002)] and M (p, q) [Gail and Mantel (1977), Diaconis and Gangolli (1995)] allowing nonregular margins; however, it appears that all are exponential in the size of the matrix, even for bounded margins. While in this work we are concerned solely with exact results, we note that many useful approximations for N (p, q) and M (p, q) (in the general case) have been found, as well as approximate sampling algorithms [Holmes and Jones (1996), Chen et al (2005), Greenhill, McKay and Table 3 Partially-filled matrix 0 0 1 1 0 1 0 1 0 1 0 1 0 6 6 4 3 1 1 1 0 0 1 1 1 1 2 2 2 3 3 3 3…”

On the Largest Size of an Antichain in the Bruhat Order for

“…[18], in the The On-Line Encyclopedia of Integer Sequences. We observe also that computing a closed manageable formula for such sequence is a still open problem which looks quite hard (cf., e.g., [1,6,11,12,13,14,15,19,20] and the references therein for some partial results).…”