New Generalized Sub Class of Cyclic-Goppa Code

Size: px

Start display at page:

Download "New Generalized Sub Class of Cyclic-Goppa Code"

Shanon Jackson
5 years ago
Views:

1 International Journal of Contemporary Mathematical Sciences Vol., 206, no. 7, HIKARI Ltd, New Generalized Sub Class of Cyclic-Goppa Code Rajesh Kumar Narula Department of Mathematics I.K. Gujral Punjab Technical University Jalandhar (Punjab), India O. P. Vinocha Department Mathematics Tantia University Sri Ganganagar (Rajastahn), India Ajay Kumar Department of Mathematics I.K. Gujral Punjab Technical University Jalandhar (Punjab), India Copyright c 206 Rajesh Kumar Narula, O. P. Vinocha and Ajay Kumar. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Cyclic codes can be viewed as ethics in an excess class of polynomials. Moreover, it is a challenging issue to construct Sub class cyclic Goppa or related Codes.Tzeng and Zimmerman proposed an idea that multiple error correcting Goppa codes can be converted in to cyclic and reversible Goppa codes by adding extra parity check. In this correspondence, we extended the idea given by Tzeng and Zimmermann and construct new generalized sub classes of cyclic Goppa codes.it is observed code (L, G) contains subclass of all cyclic codes which are reversible and cyclic Keywords: Goppa Codes, Cyclic Goppa Codes, Subclass, Reversible Codes

2 334 Rajesh Kumar Narula, O. P. Vinocha and Ajay Kumar Introduction For many years, the Goppa codes have been interesting in the field of coding theory, as these codes are generally non cyclic; however, class of Goppa codes is cyclic and is known as BCH codes.. By extending an overall parity check the double error-correcting binary Goppa codes, with defined Goppa polynomial and location sets, can become cyclic as proved by Berlekamp and Moreno. After that Tzeng and Zimmermann shown that a number of multiple error correcting Goppa codes converted in to cyclic when these codes are extended by an overall parity check. They also proved that these codes are reversible. In this paper we extended the idea proposed by Tzeng and Zimmermann and constructed a new subclass of Goppa code with extended parity check matrix and proved that these codes are reversible and cyclic. We give the following brief detail about the Goppa codes. The following polynomial with degree t over the extended field GF (q m ) g(x) = g 0 + g x + g 2 x g t x t () represents Goppa polynomial of the Goppa codeγ(l, g(z) L(X) = W, W 2...W n GF (q m (2) Such that g(α n ) 0 for allα n L with a vector C = c, c 2...c n over GF (q)we associate the function n c i R c (x) = (3) x w n n= In which is the unique polynomial with (x w i ) (modg(x)) x w n x w n The Goppa codeγ(l, g(z)) consists of all vector c such that R c (x) 0(modg(x)) Argument. The code Γ(L, g(z)) represents a Goppa code of length n over GF(q)with added further conditions of dimension k i jt and the minimum distance of the code satisfies d t + Proof we would like to prove first the lower bound on the dimension as pointed earlier can be represented as a polynomial P i (x)modulog(x) x w n = P n + P n2 x +...P nt x t (4) x w n Thus we can rewrite equation as i n= c np n (x) 0mod(g(x)) i c n P mn 0, for m t (5) n= clearly (L, g(z))termed as t linear equation over the GF(x) which convert to mt linear equation over the field GF (q)

3 New generalized sub class of cyclic-goppa code 335 Parity check matrix of the Goppa code In the proof of above Theorem. we proved that c proposes a codeword if and only if i c n P mn 0, for m t (6) n= with P mn such that = P n + P n2 x +...P nt x t The parity check x w n matrix H satisfies ch t = 0 p...p n... p t...p nt To determine the factor we rewrite P n (X) (x ω n ) g(x) g(x n x ω n.g(x) (7) This can be checked by multiplication with (x ω n ) (x ω n ). g(x) g(x n.g(x) = g(x)g(α n ) + mod(g(x)) We now x ω n define h i = g(ω n ) and recall that g(x) = g 0 + g x... + g n x t P n (x) = g t(x t ωn)... t + g (x ω n ) h n x ω n if we now substitute P n (x) = P n + P n2 x P nt x t, we find the following expression for P mn P n = (g t ωn t + g t ω t g 2 ω n + g n )h n P n2 = (g t ωn t 2 + g t ω t g 2 )h n P n(t ) = (g t ω n + g t )h n P nt = (g t )h n after calculating we find that CXY for g t g t g t 2... g 0 g t g t... g 2 C = 0 0 g t...g 3 : : :... : g t X = 2...αn t 2...αn t 2 : :... : α 2...α n... α t α t α t 2 α t 2 α

4 336 Rajesh Kumar Narula, O. P. Vinocha and Ajay Kumar h h Y = 0 0 h : : :... : h n obviously C is invertible and in addition code can be represented by another parity check matrix that is α t α t 2 h n h n h α2 t h 2...αn t h α2 t 2 h 2...αn t 2 : :... : α h α 2 h 2...α n h n h h 2...h n A.The following matrix H E is known as the parity check matrix of the classical Goppa code with added parity check. H E = ( H(L,G) ) 0.. Where H E represents as the parity check matrix of a separable Goppa code of G(x) with degree two and location set LGF (q m ) B. The H P C also represents parity check matrix of the classical cyclic Goppa code ( ) H(L,G) H PC =.. 2 Reversible Goppa Codes Let L = ω, ω 2...ω n, ω n+...ω 2n be a subset of GF (q m ), and Let g(z) be a polynomial in z over the above said field GF (q m ) with a condition that no root of g(z) is in L and also degree of g(z) is less than 2n. A Goppa code is hence defined as the set of n-tuplesx, x 2...x n, x n+, x n+2..x 2n with x i GF (q) such that n x 2i i= 0modg(z)where g(z) z ω 2i g(z) = (z θ ) q + (z θ 2 ) q (z θ n ) qn + (z θ n+ ) q n+ +...(z θ 2n ) q 2n (8)

5 New generalized sub class of cyclic-goppa code 337 (θ ω ) (θ ω 2 )...(θ ω n ) (θ ω n+ ) (θ ω n+2 )...(θ ω 2n ) (θ ω ) 2 (θ ω 2 ) 2...(θ ω n ) 2 (θ ω n+ ) 2 (θ ω n+2 ) 2...(θ ω 2n ) 2 : :... : : :... : (θ ω ) q (θ ω 2 ) q...(θ ω n ) q (θ ω n+ ) q (θ ω n+2 ) q...(θ ω 2n ) q (θ 2 ω ) (θ 2 ω 2 )...(θ 2 ω n ) (θ 2 ω n+ ) (θ 2 ω n+2 )...(θ 2 ω 2n ) (θ 2 ω ) 2 (θ 2 ω 2 ) 2...(θ 2 ω n ) 2 (θ 2 ω n+ ) 2 (θ 2 ω n+2 ) 2...(θ 2 ω 2n ) 2 : :... : : :... : (θ 2 ω ) q (θ 2 ω 2 ) q...(θ 2 ω n ) q (θ 2 ω n+ ) q (θ 2 ω n+2 ) q...(θ 2 ω 2n ) q : :... : : :... : (θ n ω ) (θ n ω 2 )...(θ n ω n ) (θ n ω n+ ) (θ n ω n+2 )...(θ n ω 2n ) (θ n ω ) 2 (θ n ω 2 ) 2...(θ n ω n ) 2 (θ n ω n+ ) 2 (θ n ω n+2 ) 2...(θ n ω 2n ) 2 : :... : : :... : (θ n ω ) q (θ n ω 2 ) q...(θ n ω n ) q (θ n ω n+ ) q (θ n ω n+2 ) q...(θ n ω 2n ) q We consider a class of Goppa code with g(z) (θ, θ 2 ) GF (q m ) and L = GF (q m ) (θ, θ 2 ) the polynomial (z θ )(z θ 2 ) is irreducible over the field GF (q m ) with added condition that θ 2 = θ qm and θ = θ qm 2 and n + n = q m therefore the parity check matix of above considered Goppa code is (θ ω ) (θ ω 2 )...(θ ω n ) (θ ω n+ ) (θ ω n+2 )...(θ ω 2n ) (θ ω ) 2 (θ ω 2 ) 2...(θ ω n ) 2 (θ ω n+ ) 2 (θ ω n+2 ) 2...(θ ω 2n ) 2 : :... : : :... : (θ ω ) q (θ ω 2 ) q...(θ ω n ) q (θ ω n+ ) q (θ ω n+2 ) q...(θ ω 2n ) q (θ 2 ω ) (θ 2 ω 2 )...(θ 2 ω n ) (θ 2 ω n+ ) (θ 2 ω n+2 )...(θ 2 ω 2n ) (θ 2 ω ) 2 (θ 2 ω 2 ) 2...(θ 2 ω n ) 2 (θ 2 ω n+ ) 2 (θ 2 ω n+2 ) 2...(θ 2 ω 2n ) 2 : :... : : :... : (θ 2 ω ) q (θ 2 ω 2 ) q...(θ 2 ω n ) q (θ 2 ω n+ ) q (θ 2 ω n+2 ) q...(θ 2 ω 2n ) q we should be noted that for each i,j (i, j) 2n and if (θ ω ) = (θ 2 ω j ) θ ω j = (θ 2 ω i ) (θ ω j ) d = [(θ 2 ω i )] d forthevaluesd = 2, 3...q Therefore the considered two cases we have θ + θ 2 GF (q m ) for each ω j L there must exist ω i = θ + θ 2 ω j LThese codes becomes reversible if we ordered them as ω 2n+ j = θ + θ 2 ω i 3 Extended Reversible Goppa Codes we extended the considered Goppa codes by adding overall parity check and the new extended Parity check matix denoted as H E is (θ ω ) (θ ω 2 )...(θ ω n ) (θ ω n+ ) (θ ω n+2 )...(θ ω 2n ) 0 (θ ω ) 2 (θ ω 2 ) 2...(θ ω n ) 2 (θ ω n+ ) 2 (θ ω n+2 ) 2...(θ ω 2n ) 2 : :... : : :... : H E = 0 (θ ω ) q (θ ω 2 ) q...(θ ω n ) q (θ ω n+ ) q (θ ω n+2 ) q...(θ ω 2n ) q 0 (θ 2 ω ) (θ 2 ω 2 )...(θ 2 ω n ) (θ 2 ω n+ ) (θ 2 ω n+2 )...(θ 2 ω 2n ) 0 (θ 2 ω ) 2 (θ 2 ω 2 ) 2...(θ 2 ω n ) 2 (θ 2 ω n+ ) 2 (θ 2 ω n+2 ) 2...(θ 2 ω 2n ) 2 : :... : : :... : 0 (θ 2 ω ) q (θ 2 ω 2 ) q...(θ 2 ω n ) q (θ 2 ω n+ ) q (θ 2 ω n+2 ) q...(θ 2 ω 2n ) q

6 338 Rajesh Kumar Narula, O. P. Vinocha and Ajay Kumar To prove H E represents a cyclic code we have to conclude that above said matrix is equivalent to the proposed matrix i.e... + θ 2 θ + θ 2 θ... + θ 2 θ θ ω θ ω 2 θ ω n ( + θ 2 θ ) 2 ( + θ 2 θ ) 2...( + θ 2 θ ) 2 θ ω θ ω 2 θ ω n ( + θ 2 θ ) n ( + θ 2 θ ) n...( + θ 2 θ ) n θ ω θ ω 2 θ ω n + θ 2 θ + θ 2 θ... + θ 2 θ θ ω θ ω 2 θ ω n ( θ 2 θ ) 2 ( θ 2 θ ) 2...( θ 2 θ ) 2 θ 2 ω θ 2 ω 2 θ 2 ω n ( θ 2 θ ) n ( θ 2 θ ) n...( θ 2 θ ) n θ 2 ω θ 2 ω 2 θ 2 ω n + θ 2 θ = θ 2 ω d θ ω d θ ω d θ 2 θ = θ ω d θ ω d θ 2 ω d = ( θ 2 θ ) θ ω d further θ 2 ω i θ 2 ω j for i j θ ω i θ ω j conjugates over GF (q m ) we get ( θ 2 ω d θ ω d ) qm + = ( θqm 2 ω qm d θ qm ω qm d )( θ 2 ω d θ ω d ) = ( θ ω d )( θ 2 ω d ) θ 2 ω d θ ω d = Suppose N= n(n+) and then N = q m have definitely N- different Nth roots of unity which are not equal to one. Consequently, the location set is expressed as, θ 2 ω d = ρ d θ ω d therefore ω d + ω n+ d = θ 2 θ ρ d + θ 2 θ ρ n+ d ρ d ρ n+ d = θ 2 θ ρ d + θ 2 θ ρ d ρ d ρ d

7 New generalized sub class of cyclic-goppa code 339 = θ 2 θ ρ d + θ θ 2 ρ d ρ d ρ d = θ + θ 2 Thus, we follow the H E ρ ρ 2...ρ N ρ (N+)...ρ (N+N) ρ 2 (ρ 2 ) 2...(ρ 2 ) N (ρ 2 ) (N+)...(ρ 2 ) (N+N) : : :... : :... : ρ d (ρ d ) d...(ρ d ) N (ρ d ) (N+)...(ρ d ) (N+N) ρ ρ 2...ρ (N ) ρ (N+)+...ρ (N+N)+ ρ 2 (ρ 2 ) 2...(ρ 2 ) N (ρ 2 ) (N+)...(ρ 2 ) (N+N) : : :... : :... : ρ d (ρ d ) d...(ρ d ) N (ρ d ) (N+)...(ρ d ) (N+N) Example The Goppa code with L = GF (2 4 ),let g(z) = (z θ)(z θ 4 ) as θ is a primitive in GF (2 4 ) and suppose we ordered the location set L such as θ i = θ3 θθ i θ i for i 4 letg(z) = [(z θ)(z θ 3 )] 4 then we get (θ θ 2 ) (θ 0)...(θ θ ) (θ θ 2 ) 2 (θ 0) 2...(θ θ ) 2 (θ θ 2 ) 3 (θ 0) 3...(θ θ ) 3 (θ θ 2 ) 4 (θ 0) 4...(θ θ ) 4 (θ 3 θ 2 ) (θ 3 0)...(θ 3 θ ) (θ 3 θ 2 ) 2 (θ 3 0) 2...(θ 3 θ ) 2 (θ 3 θ 2 ) 3 (θ 3 0) 3...(θ 3 θ ) 3 (θ 3 θ 2 ) 4 (θ 3 0) 4...(θ 3 θ ) 4 θ 0 θ 4...θ 9 θ 5 θ 3...θ 3 θ 2...θ 2 θ 4 θ...θ θ 9 θ...θ 0 θ 3 θ 9...θ 5 θ 2 θ 6... θ 0 θ 5...θ 2

8 340 Rajesh Kumar Narula, O. P. Vinocha and Ajay Kumar... ( + θ 9 θ 0 )...( + θ 9 θ 9 ) ( + θ 9 θ 0 ) 2...( + θ 9 θ 9 ) 2 ( + θ 9 θ 0 ) 3...( + θ 9 θ 9 ) 3 H E = ( + θ 9 θ 0 ) 4...( + θ 9 θ 9 ) 4 ( θ 9 θ 0 )...( θ 9 θ 9 ) ( θ 9 θ 0 ) 2...( θ 9 θ 9 ) 2 ( θ 9 θ 0 ) 3...( θ 9 θ 9 ) 3 ( θ 9 θ 0 ) 4...( θ 9 θ 9 ) 4... θ...θ 4 θ 2...(θ 4 ) 2 θ 3...(θ 4 ) 3 H E = θ 4...(θ 4 ) 4 θ...(θ 4 ) θ 2...(θ 4 ) 2 θ 3...(θ 4 ) 3 θ 4...(θ 4 ) 4 which defines a reversible (6, 2) cyclic code generated by g(x) = (x+)(x 4 +x+)(x 4 +x 3 +)(x 4 +x 3 +x 2 +x+)(x 5 +x 4 +x 3 +x 2 +x+) 4 Conclusion Goppa codes and a set of extended reversible Goppa codes have been formulated and a new extended subclass of cyclic and reversible codes is presented in this study, resulting in the representation of extended reversible Goppa codes. An additional example given above is considered,where g(x) = (x + )(x 4 + x + )(x 4 + x 3 + )(x 4 + x 3 + x 2 + x + )(x 5 + x 4 + x 3 + x 2 + x + )is termed as a reversible code with Goppa polynomial of the degree. This work arise a certain number of new research problems. In future, we are going to enhance a moderated extended reversible code for a generalized cyclic Goppa code. References [] Sergey V. Bezzateev and Natalia A. Shekhunova, Subclass of cyclic Goppa codes, IEEE Transactions on Information Theory, 59 (203), no.,

9 New generalized sub class of cyclic-goppa code [2] Elwyn R. Berlekamp, Goppa codes, IEEE Transactions on Information Theory, 9 (973), no. 5, [3] Sergei V. Bezzateev and Natalia A. Shekhunova, Subclass of binary Goppa codes with minimal distance equal to the design distance, IEEE Transactions on Information Theory, 4 (995), no. 2, [4] Sergey Bezzateev and Natalia Shekhunova, Chain of separable binary Goppa codes and their minimal distance, IEEE Transactions on Information Theory, 54 (2007), no. 2, [5] Sergei V. Bezzateev and Natalia A. Shekhunova, One generalization of Goppa codes, Proceedings IEEE International Symposium on Information Theory, (997). [6] K. K Tzeng and K. Zimmermann, On Extendind Goppa Codes to Cyclic Codes, IEEE Transactions on Information Theory, 2 (975), Received: July, 206; Published: July 30, 206

CYCLIC SEPARABLE GOPPA CODES

CYCLIC SEPARABLE GOPPA CODES bsv@aanet.ru sna@delfa.net Saint Petersburg State University of Aerospace Instrumentation Russia Algebraic and Combinatorial Coding Theory June 15-21, 2012 Pomorie, Bulgaria Outline Overview of previous