Global Journal of Science Frontier Research, F: Mathematics and Decision Science, Volume 21 Issue 4

Maximum Distance Separable Codes to Order By explicit row selection, the process is as follows. Denote the rows of U in order by h e 0 , e 1 , . . . , e n − 1 i and the columns of V in order by h f 0 , f 1 , . . . , f n − 1 i . Then   e 0 e 1 ... e n − 1   ( f 0 , f 1 , . . . , f n − 1 ) = I n . From this it is seen that e i f i = 1 , e i f j = 0 , i 6 = j . Thus if G =   e i 1 e i 2 ... e i r   (for distinct e i k ) and H T = ( f j 1 , f j 2 , . . . , f j n − r ) where { j 1 , j 2 , . . . , j n − r } = { 0 , 1 , . . . , n − 1 } / { i 1 , i 2 , . . . , i r } . Then GH T = 0 r × ( n − r ) . Both G and H have full rank. When the first r rows chosen this gives   e 0 e 1 ... e r   ( f 0 , f n − 1 , f n − 2 , . . . , f n − r ) = 0 r × ( n − r ) for the code system expressing the generator and check matrices. When the rows are chosen from Vandermonde/Fourier matrices and taken in arithmetic sequence with arithmetic difference k satisfying gcd( n, k ) = 1 then MDS codes are obtained. In particular when k = 1, that is when the rows are taken consecutively, MDS codes are obtained. This follows from results in [6] and these are explicitly recalled in Theorems 2.1, 2.2 below. The n × n Vandermonde matrix V ( x 1 , x 2 , . . . , x n ) is defined by V = V ( x 1 , x 2 , . . . , x n ) =   1 1 . . . 1 x 1 x 2 . . . x n ... ... ... ... x n − 1 1 x n − 1 2 . . . x n − 1 n   As is well known, the determinant of V is Q i<j ( x i − x j ). Thus det( V ) 6 = 0 if and only the x i are distinct. A primitive n th root of unity ω in a field F is an element ω satisfying ω n = 1 F but ω i 6 = 1 F , 1 ≤ i < n . Often 1 F is written simply as 1 when the field is clearly understood. The field GF ( q ) (where q is necessarily a power of a prime) contains a primitive ( q − 1) root of unity, see [1, 18] or any book on field theory, and such a root is referred to as a primitive element in the field GF ( q ). Thus also the field GF ( q ) contains a primitive n th roots of unity for any n/ ( q − 1). A Fourier n × n matrix over F is a special type of Vandermonde matrix in which x i = ω i − 1 and ω is a primitive n th root of unity in F . Thus: F n =   1 1 1 . . . 1 1 ω ω 2 . . . ω n − 1 1 ω 2 ω 4 . . . ω 2( n − 1) ... ... ... . . . ... 1 ω n − 1 ω 2( n − 1) . . . ω ( n − 1)( n − 1)   is a Fourier matrix over F where ω is a primitive n th root of unity in F . 1 Global Journal of Science Frontier Research Volume XXI Issue IV Year 2021 3 ( F ) © 2021 Global Journals Version I particular this gives AD = 0 r × ( n − r ) . The matrices have full rank. Thus with A as the generating matrix of an ( n, r ) code it is seen that D T is the check matrix of the code. b) Vandermonde/Fourier matrices 6. Ted Hurley and Donny Hurley, “ Coding theory: the unit-derived methodology ” , Int. J. Information and Coding Theory, Vol. 5, no.1, 55-80, 2018. R ef