Cyclic permutation

Abstract: A file protection scheme for fixed content in a distributed data archive uses computations that leverage permutation operators of a cyclic code. In an illustrative embodiment, an N+K coding technique is described for use to protect data that is being distributed in a redundant array of independent nodes (RAIN). The data itself may be of any type, and it may also include system metadata. According to the invention, the data to be distributed is encoded by a dispersal operation that uses a group of permutation ring operators. In a preferred embodiment, the dispersal operation is carried out using a matrix of the form [I N — C] where I N is an n×n identity sub-matrix and C is a k×n sub-matrix of code blocks. The identity sub-matrix is used to preserve the data blocks intact. The sub-matrix C preferably comprises a set of permutation ring operators that are used to generate the code blocks. The operators are preferably superpositions that are selected from a group ring of a permutation group with base ring Z 2 .

Abstract: real procedure PCpolynomial (x, n, a); integer n; real x, a; comment PCpolynomial computes values of the Poisson-Charlier polynomial p~(x) defined by L. Carlitz, Characterization of certain sequences of orthogonal polynomials, Portugaliae Mathematica 20 (1961), 43-46: () (:) p~(x) a~12(n~)_ll~ ~ (_1).~ n =. r! a ~ r=0 r • In this algorithm u stands for the successive terms of the summa-tion, s stands for the sum of these terms and all other symbols possess evident meanings. Clearly each term of the summation is obtained from the preceding one by the indicated multiplication ; c:=l; for] := 1 step 1 untilndoc := c X j; for j := 0 step 1 until n-1 do begin u :=-u X (n-j) X (x-j)/(a X (]-~-1)); s := s + u end; PCpolynomial := sqrt(a Tn/c) X s end PCpolynomial integer array a; begin comment SHUFFLE applies a random permutation to the sequence a[i] where i = 1, 2, ... , n. The procedure random is supposed to supply a random element from a large population of real numbers uniformly distributed over the open unit interval 0 < r < 1. The array a is declared to be integer but actually it suffices for its type to agree with that of the variable b (in the procedure body); integer i, j; real b; for i := n step-1 until 2 do begin 3" := entier (i X random-~ 1); b := a[i]; a[i] := a~']; a[]] := b end loop i end SHUFFLE Note. Numbers in brackets following Algorithm titles indicate the subject category for the algorithm, based on the Modified SHARE Classification listing given in the 1Viarch, 1964 issue of the Communications of the ACM. The Romberg integration algorithm has been used with great success by many groups [1, 2], and appears to be among the most generally reliable quadrature methods available. It is, therefore, worth pointing out that it is not entirely foolproof, and that a significant class of integrands exists for which the extrapolated values are poorer estimates of the integral than the corresponding trapezoidal sums. The validity of the Romberg procedure depends upon the possibility of expanding the error of the trapezoidal rule in powers of h 2, where h is the stepsize. One expansion of this type is the Euler-Maclaurin sum formula. An alternative expression may be obtained from the Fourier series expansion. The coefficients of h 2" …