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

Maximum Distance Separable Codes to Order In characteristic p the rates r n attainable require p 6 | n so that the Fourier matrix n × n can be constructed in characteristic p . This is not a great restriction. For any given fraction R and any given ǫ > 0 there exists a fraction with numerator not divisible by p between R and R + ǫ . The details are omitted. For example suppose in characteristic 2 the rate required is 3 4 and ǫ > 0 is given. Say 1 32 < ǫ and then need a fraction of the required type between 3 4 and 3 4 + 1 32 = 25 32 . Now 24 31 will do and we can proceed with this fraction to construct the codes over characteristic 2; the Fourier 31 × 31 matrix exists over GF (2 5 ). Arithmetic in prime fields is particularly nice. Here we develop a method for constructing series of MDS codes over prime fields. Suppose a rate R is required, 0 < R < 1. Let p be a prime and consider the field GF ( p ) = Z p . This has an element of order ( p − 1) and thus construct the Fourier ( p − 1) × ( p − 1) matrix F p − 1 over GF ( p ) = Z p . For this it is required to find a primitive ( p − 1) root of unity in GF ( p ) = Z p . 2 Let r = ⌊ ( p − 1) ∗ R ⌋ . Now p must be large enough so that r ≥ 1. Form the ( p − 1 , r ) MDS code over F p − 1 . This has rate close to R . Let { p 1 , p 2 , . . . , p i , . . . } be an infinite increasing set of primes such that ( p 1 − 1) ∗ R ≥ 1 in which case ( p i − 1) ∗ R ≥ 1 for each i . Form the Fourier ( p i − 1) × ( p i − 1) matrix over GF ( p i ). Let r i = ⌊ ( p i − 1) ∗ R ⌋ . Form the ( p i − 1 , r i ) MDS code over GF ( p i ). The ratio of the distance to the length is p i − 1 − r i +1 p i − 1 = 1 − r i p i − 1 + 1 p i − 1 . Now as i → ∞ this ratio approaches (1 − R ). Sample Let { p 1 , p 2 , . . . } be the primes of the form (4 n + 1) and let R = 3 4 . Now p 1 = 5 , p 2 = 13 , p 3 = 17 , . . . . Let r i = ( p i − 1) ∗ R = 4 ∗ j ∗ 3 4 = j ∗ 3 for some j . Form the ( p i − 1 , r i ) code from Fourier ( p i − 1) × ( p i − 1) matrix over GF ( p i ). Get codes { (4 , 3 , 2) , (12 , 9 , 4) , (16 , 12 , 5) , (28 , 21 , 8) , (36 , 27 , 10) , . . . } over, respectively, the following fields { GF (5) , GF (13) , GF (17) , GF (29) , GF (37) , . . . } . Here is an example of MDS codes in GF (13) = Z 13 . A primitive element in GF (13) is ω = (2 mod 13). The Fourier 12 × 12 matrix with this ω as the element of order 12 in GF (13) = Z 13 is: F 12 =   1 1 1 1 1 1 1 1 1 1 1 1 1 2 4 8 3 6 12 11 9 5 10 7 1 4 3 12 9 10 1 4 3 12 9 10 1 8 12 5 1 8 12 5 1 8 12 5 1 3 9 1 3 9 1 3 9 1 3 9 1 6 10 8 9 2 12 7 3 5 4 11 1 12 1 12 1 12 1 12 1 12 1 12 1 11 4 5 3 7 12 2 9 8 10 6 1 9 3 1 9 3 1 9 3 1 9 3 1 5 12 8 1 5 12 8 1 5 12 8 1 10 9 12 3 4 1 10 9 12 3 4 1 7 10 5 9 11 12 6 3 8 4 2   Let the rows of F 12 in order be denoted by { e 0 , e 1 , . . . , e 11 } . Various MDS codes over GF (13) may be constructed from F 12 . Two of the (12 , 6 , 7) codes are as follows: K =   1 1 1 1 1 1 1 1 1 1 1 1 1 2 4 8 3 6 12 11 9 5 10 7 1 4 3 12 9 10 1 4 3 12 9 10 1 8 12 5 1 8 12 5 1 8 12 5 1 3 9 1 3 9 1 3 9 1 3 9 1 6 10 8 9 2 12 7 3 5 4 11   , L =   1 2 4 8 3 6 12 11 9 5 10 7 1 12 1 12 1 12 1 12 1 12 1 12 1 7 10 5 9 11 12 6 3 8 4 2 1 3 9 1 3 9 1 3 9 1 3 9 1 5 12 8 1 5 12 8 1 5 12 8 1 4 3 12 9 10 1 4 3 12 9 10   1 Global Journal of Science Frontier Research Volume XXI Issue IV Year 2021 7 ( F ) © 2021 Global Journals Version I i. Note e) Infinite series in prime fields with given rate f) Sample of the workings 2 It seems there is no known algorithm for finding a generator of that is substantially better than a brute force method - see Keith Conrad’s notes [4]. Note however there are precisely generators. ( Z p / { 0 } ( φ ( p − ( N otes