in this paper, we consider a class of column-weight two quasi-cyclic low-density paritycheck codes in which the girth can be large enough, as an arbitrary multiple of 8. then we devote a convolutional form to these codes, such that their generator matrix can be obtained by elementary row and column operations on the parity-check matrix. finally, we show that the free distance of the convolution...

‎A rank equality is established for the sum of finitely many tripotent matrices via elementary block matrix operations‎. ‎Moreover‎, ‎by using this equality and Theorems 8 and 10 in [Chen M‎. ‎and et al‎. ‎On the open problem related to rank equalities for the sum of finitely many idempotent matrices and its applications‎, ‎The Scientific World Journal 2014 (2014)‎, ‎Article ID 702413‎, ‎7 page...

C Elementary and special functions (search also class L5 ) C1 Integer-valued functions (e.g., factorial, binomial coefficient, permutations, combinations, floor, ceiling) C06GXFP Factorizes a positive integer n as n = n1 × n2. This routine may be used in conjunction with C06MCFP D Linear Algebra D1 Elementary vector and matrix operations D1a Elementary vector operations D1a1 Set to constant D1a...

Consider a transformed linear mixed model (TLMM) obtained pre-multiplying (LMM) M : y = Zα + Rγ e by given matrix. This work concerns the problem of equalities predictors under considered two LMMs general assumptions. We characterize between best unbiased (BLUPs) LMM and its TLMM using various rank formulas block matrices elementary matrix operations.

8 Elementary operations and the rank of a matrix 25 8.1 Gauss transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 8.2 Elementary row operations . . . . . . . . . . . . . . . . . . . . . . . . . 26 8.3 The augmented matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 8.4 Row reducing a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 8.5 Elem...

In this paper, we consider a class of column-weight two quasi-cyclic low-density parity check codes in which the girth can be large enough, as an arbitrary multiple of 8. Then we devote a convolutional form to these codes, such that their generator matrix can be obtained by elementary row and column operations on the parity-check matrix. Finally, we show that the free distance of the convolutio...

A quantum compiler is a software program for decomposing (“compiling”) an arbitrary unitary matrix into a sequence of elementary operations (SEO). Coppersmith showed that the NB-bit Discrete Fourier Transform matrix UFT can be decomposed in a very efficient way, as a sequence of order(NB ) elementary operations. Can a quantum compiler that doesn’t know a priori about Coppersmith’s decomposition...

Efficiently manipulating and operating on block matrices can be beneficial in many applications, among others those involving iteratively solving nonlinear systems. These types of problems consist of repeatedly assembling and solving sparse linear systems. In the case of very large systems, without a careful manipulation of the corresponding matrices, solving can become very time consuming. Thi...

A linear mixed model ($\LMM$) $\M :\yy = \mxX\BETA + \mxZ\uu \EPS $ with general assumptions and its transformed $\T:\mxT\yy \mxT\mxX\BETA \mxT\mxZ\uu \mxT\EPS are considered. This work concerns the comparison problem of predictors under $\M$ $\T$. Our aim is to establish equality relations between best unbiased ($\BLUP$s) unknown vectors two $\LMM$s $\T$ through their covariance matrices by us...