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

Maximum Distance Separable Codes to Order In addition (5 mod 257) or (7 mod 257) could be used to generate the Fourier 256 × 256 matrix over GF (257) = Z 257 ; indeed there exist φ (256) = 128 generators that could be used to generate the Fourier matrix. Note that (256 , 240 , 17) and (256 , 224 , 23) codes over GF (257) are constructed as well as other rate codes. These particular ones could be compared to the Reed-Solomon (255 , 239 , 17) and (255 , 223 , 23) codes which are in practical use; the ones from GF (257) perform better and faster. There is a much bigger choice for rate and error-correcting capability. Bigger primes could also be used. Taking p = 509 gives (508 , r ) MDS codes for any 1 < r < 508. Thus for example (508 , 486 , 23) MDS codes over GF (509) = Z 509 are constructed. The method allows the construction of φ (256) = 128 such MDS (256 , r ) codes with different gener- ators for the Fourier matrix. For larger primes the number that could be used for the construction of the Fourier matrix is substantial and cryptographic methods could be devised from such con- siderations. For example for the prime p = 2 31 − 1 the Fourier ( p − 1) × ( p − 1) matrix exists over GF ( p ) and φ ( p − 1) = 534600000 elements could be used to generate the Fourier matrix. 6. The p can be very large and the arithmetic is still doable. For example p = 10009 allows the construction of (10008 , r ) MDS codes over GF (10009) = Z 10009 . If 100 errors are required to be corrected the scheme supplies (10008 , 9808 , 201) MDS codes over GF (10009) = Z 10009 which have large rate ≈ . 98 and can correct 100 errors. The arithmetic is modular arithmetic. The order of ω = (11 mod 10009) is 10008 so this ω could be used to generate the Fourier 10008 × 10008 matrix over GF (10009) = Z 10009 ; indeed there are φ (10008) = 3312 different elements in GF (10009) = Z 10009 that could be used to generate the Fourier 10008 × 10008 matrix. 7. General p . Then MDS codes of the form ( p − 1 , p − 3 , 3) , ( p − 1 , p − 5 , 5) , ( p − 1 , p − 7 , 7) , . . ., ( p − 1 , p − (2 i +1) , 2 i +1) , . . . , ( p − 1 , 2 , p − 2) are constructed which can respectively correct { 1 , 2 , 3 , . . . , i, . . . p − 3 2 } errors are constructed. A primitive modular element (of order ( p − 1)) is obtained in GF ( p ) = Z p with which to construct the Fourier matrix; as already noted it seems a brute force method for obtaining such seems to be as good as any. Suppose n is given and it is required to find finite fields over which a Fourier n × n matrix exists. The following argument is essentially taken from [6]. It is included for clarity and completeness and is necessary for deciding on the relevant fields to be used in cases. Note first of all that the field must have characteristic which does not divide n in order for the Fourier n × n matrix to exist over the field. There exists a finite field of characteristic p containing an n th root of unity for given n if and only if p 6 | n . Let p be a prime which does not divide n . Hence p φ ( n ) ≡ 1 mod n by Euler’s theorem where φ denotes the Euler φ function. More specifically let β be the least positive integer such that p β ≡ 1 mod n . Consider GF ( p β ). Let δ be a primitive element in GF ( p β ). Then δ has order ( p β − 1) in GF ( p β ) and ( p β − 1) = sn for some s . Thus ω = δ s has order n in GF ( p β ). On the other hand if p/n then n = 0 in a field of characteristic p and so no n th root of unity can exist in the field. The proof is constructive. Let n be given and p 6 | n . Let β be the least power such that p β ≡ 1 mod n ; it is known that p φ ( n ) ≡ 1 mod n and thus β is a divisor of φ ( n ). Then the Fourier n × n matrix over GF ( p β ) exists. Sample Suppose n = 52. The prime divisors of n are 2 , 13 so take any other prime p and then there is a field of characteristic p which contains a 52 nd root of unity. For example take p = 3. Know 3 φ (52) ≡ 1 mod 52 and φ (52) = 24 but indeed 3 6 ≡ 1 mod 52. Thus the field GF (3 6 ) contains a primitive 52 nd root of unity and the Fourier 52 × 52 matrix exists in GF (3 6 ). Also 5 4 ≡ 1 mod 52, and so GF (5 4 ) can be used. Now 5 4 = 625 < 729 = 3 6 so GF (5 4 ) is a smaller field with which to work. © 2021 Global Journals 1 Global Journal of Science Frontier Research Volume XXI Issue IV Year 2021 10 ( F ) Version I i) The fields Proposition 3.1 Proof: 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