The True Minimum Distance of Some Narrow-Sense BCH-Codes of Length 255

Using equivalent codes it is shown that the BCH-bound of the following narrow-sense BCH-codes already yields the true minimum distance: 255,87,53, 255,107,45, 255,115,43, 255,123,39, 255,131,37, 255,147,29, 255,163,25, 255,179,21. For the remaining two narrow-sense BCH-codes of length 255 in the book of F.J. MacWilliams and N.J.A. Sloane The theory of error-correcting codes (1977) page 261, figure 9.1, whose true minimum distance is still unknown, upper bounds for the minimum distance are given which differ by at most two from the corresponding BCH-bounds

Block Sieving Algorithms

Quite similiar to the Sieve of Erastosthenes, the best-known general algorithms for factoring large numbers today are memory-bounded processes. We develop three variations of the sieving phase and discuss them in detail. The fastest modification is tailored to RISC processors and therefore especially suited for modern workstations and massively parallel supercomputers. For a 116 decimal digit composite number we achieved a speedup greater than two on an IBM RS/6000 250 workstation

Factoring Integers above 100 Digits using Hypercube MPQS

In this paper we report on further progress with the factorisation of integers using the MPQS algorithm on hypercubes and a MIMD parallel computer with 1024 T-805 processors. We were able to factorise a 101 digit number from the Cunningham list using only about 65 hours computing time. We give new details about the hypercube sieve initialisation procedure and describe the structure of the factor graph that saves a significant amount of computing time. At March 3rd, we finished the factorisation of a 104 digit composite

The True Minimum Distance of Some Narrow-Sense BCH-Codes of Length 255

Using equivalent codes it is shown that the BCH-bound for the following narrowsense BCH-codes already yields the true minimum distance: [255,87,53], [255,107,45], [255,115,43], [255,123,39], [255,131,37], [255,147,29], [255,163,25], [255,179,21]. For the remaining two narrow-sense BCH-codes of length 255 in the book of F. J. MacWilliams and N. J. A. Sloane, page 261, Figure 9.1, whose true minimum distances are still unknown, upper bounds on the minimum distances are given which differ only by two from the corresponding BCH-bounds. Even 16 years after the first printing of "The Theory of Error-Correcting Codes" the table of primitive BCH-codes of length up to 255 ([3], 7 th printing, p.261) contains ten entries where the true minimum distances are unknown. We show that the BCHbound gives the true minimum distance for the following codes: [255,87,53], [255,107,45], [255,115,43], [255,123,39], [255,131,37], [255,147,29], [255,163,25], [255,179,21] by explicitly giving a check polynomia..

The True Minimum Distance of Some Narrow-Sense BCH-Codes of Length 255

Using equivalent codes it is shown that the BCH-bound of the following narrow-sense BCH-codes already yields the true minimum distance: [255,87,53], [255,107,45], [255,115,43], [255,123,39], [255,131,37], [255,147,29], [255,163,25], [255,179,21]. For the remaining two narrow-sense BCH-codes of length 255 in the book of F.J. MacWilliams and N.J.A. Sloane [The theory of error-correcting codes (1977)] page 261, figure 9.1, whose true minimum distance is still unknown, upper bounds for the minimum distance are given which differ by at most two from the corresponding BCH-bounds

The Setup Polyhedron of Series-Parallel Posets

To every linear extension L of a poset P=(P,<) we associate a {0,1}-vector x = x(L) with xe = 1 if and only if e is preceded by a jump in L or e is the first element in L. Let Q = conv{ x(L) | L in L(P) } be the convex hull of all incidence vectors of linear extensions of P. For the case of series-parallel posets we give a linear description of Q

Integer polyhedra from supermodular functions on series-parallel posets

We define a class of integer polyhedra induced by supermodular functions on series-parallel posets. We show that permutahedra and generalized polymatroids are special instances of our approach