Cyclic Codes over Some Finite Rings

Cyclic Co des o v er Some Finite Rings Mehmet ¨ Ozen, Murat G ¨ uzeltepe Department of Mathematics, Sak arya Universit y , TR54187 Sak arya, T urk ey Abstract In this pap er cyclic codes are established with resp ect to the Mannheim metric o ver some fin ite rings by using Gaussian integers and t he decod ing algorithm f or th ese codes is given. AMS Classification: 94 B05, 94B 60 Keywords: B lock codes, M annheim distance, Cyclic codes, Syndrome deco ding 1 In tro duction Mannheim metric, which was initially put for w ard by Hub er 1994 , has b een used in many pap ers so far [1, 2 , 3 , 4 , 5]. In 1994, Hub er defined the Mannheim metric and the Ma nnheim weight ov er Gaussia n integers and, event ually , he obtained the linear codes whic h can cor rect err ors of Mannheim weigh t one in [1]. Moreov er, so me of t hese codes which a re suited for quadr ature amplitu de mo dulation (QAM)-t yp e mo dulations w ere considered b y Hub er. Later, Huber transferred these co des, which he obtained by using the Mannheim metric and Gaussian in teger s , into Eisens tein- Jacobi integers in [2]. In 1997, Hub er prov ed the Ma cWilliams t heorem f or t wo-dimensional mo dulo metric ( Mannheim met- ric) [3 ]. In 2 001, Neto obtained new codes ov er Euclidean domain Q √ d , where d = − 1 , − 2 , − 3 , − 7 , − 11 , in [4] by using the Mannheim metric given in [1, 2]. In 200 4, F an a nd Gao obtained one-err or corr e c ting linea r co des by using a nov el Mannheim weight ov er finite algebraic integer rings o f cycloto mic fields [5]. In our study , the co des in [1 ] a re transferr ed in to so me finite ring by using the Mannheim metric and Gaussian integers. Since these co des are transferr ed from finite fields in to finite rings, there o ccur s some difference in deco ding pro- cess and w e a lso men tion these differences. Section I I is org anized as follows. In Prop osition 1 , the necess ary algebraic background is revealed in order to obtain cyclic co des. In Theorem 2, it is shown how to obta in cyclic co des by utilizing P rop osition 1 , 2 and 3 . In Pr op osition 4, the alge braic background, which is es sent ial fo r obtaining cyclic co de over the other finite ring s, is arrang ed a nd in Theorem 3, it is shown how to obtain cyclic co des o v er the other finit e rings. 1 2 Cyclic Co des o v er Gaussian In teger A Gaussian integer is a complex num b er whose re a l and imaginar y parts are bo th integers. Let G de no te the set of all Ga ussian in tegers and let G π denote residue cla ss of G mo dulo π , wher e π = a + ib is a Gauss ia n prime integer and p is a prime integer such tha t p = a 2 + b 2 = 4 n + 1 The mo dulo function GF ( p ) → G π is defined b y µ ( g ) = g − [ g .π ∗ p ] .π , (1) where GF ( p ) is a finite field with p elemen ts. elements. In (1), the symbo l o f [.] is rounding to the closes t integer. The rounding of Gaussia n in teg er can b e done by rounding the r eal and imaginary pa rts separ ately to the closes t integer. In view o f equation (1), G π is iso morphic to Z p , where Z p is r esidue class of the set Z of all integers mo dulo p . L et α and β b e ele ments in G π then the Mannheim weigh t of γ is defined by w M ( γ ) = | Re( γ ) | + | Im( γ ) | , where γ = α − β mod π . Since t he linear codes are linear code, the Mannheim distance betw een α and β is d M ( α, β ) = w M ( γ ) [1]. Theorem 1 If a and b ar e r elatively prime inte gers, then G = Z [ i ]/ h a + ib i ∼ = Z a 2 + b 2 [6]. Prop osition 1 L et π = a + b i b e a prime in G and let p > 2 b e prime element in Z s uch tha t p = a 2 + b 2 = 4 n + 1 . If G is a ge ner ator of G ∗ π 2 , then g φ ( p 2 ) / 4 ≡ i mo d π 2 ( or g φ ( p 2 ) / 4 ≡ − i mo d π 2 ) . Pro of. If | π | = 4 n + 1 is a pr ime in tege r in Z , the real and imagina ry par ts of π 2 are relatively primes, where the sym b o l |·| denotes modulo of a complex num b er. So, G π 2 is isomorphic to Z p 2 (See Theorem 1). If g is a gener ator of G ∗ π 2 , then g , g 2 , ..., g φ ( p 2 ) mo d π 2 constitute a reduced r esidue sy s tem. Therefore there is a p ositive in teger k as g k ≡ i mo d π 2 ( g k ≡ − i mo d π 2 , where 1 ≤ k ≤ φ ( p 2 ). Hence, w e can infer g 4 k ≡ 1 mod π 2 . Since φ ( p 2 )   4 k and 4 ≤ 4 k ≤ 4 φ ( p 2 ) , we obtain φ ( p 2 ) = k , φ ( p 2 ) = 2 k or φ ( p 2 ) = 4 k . If φ ( p 2 ) = k was equal to k o r 2 k , we should have π 2   i − 1 or π 2   2 ( π 2   − i − 1), but this would contradict the fact that   π 2   2 > 2. Prop osition 2 L et π = a + ib b e a prime in G and let p > 2 b e prime in Z such that p = a 2 + b 2 = 4 n + 1 . If g is a gener ator of G ∗ π k , the g φ ( p k ) / 4 ≡ i mo d π k or ( g φ ( p k ) / 4 ≡ − i mo d π k ). Pro of. This is immediate from P r op osition 1. Prop osition 3 L et π = a + ib b e a prime in G and let p > 2 b e prime in Z such that p = a 2 + b 2 = 4 n + 1 . If g is a gener ator of G ∗ π k and g φ ( p 2 ) / 4 ≡ i mod π 2 , then − g also b e c omes a gener ator of G ∗ π 2 such that ( − g ) φ ( p 2 ) / 4 ≡ − i mod π 2 . Pro of. g φ ( p 2 ) / 4 ≡ i mo d π 2 implies that( − g ) φ ( p 2 ) / 4 ≡ − i mod π 2 since φ ( p 2 ) = 4 n (4 n + 1 ) and n is a n odd integer. 2 Theorem 2 L et p > 2 is a prime in Z and π = a + ib is a prime in G su ch that p = a 2 + b 2 = 4 n + 1 ( a, b , n ∈ Z ) , then cyclic c o des of length φ ( p 2 ) / 4 and φ ( p 2 ) / 2 ar e gener ate d over the ring G π 2 whose t he gener ator p olynomial ar e of the fi rs t and se c ond de gr e e, r esp e ctively. Pro of. There is a n element g of G π 2 and G ∗ π 2 is genera ted by g since Z p 2 is iso- morphic to G π 2 . W e kno w that g φ ( p 2 ) / 4 ≡ i mod π 2 implies t hat ( − g ) φ ( p 2 ) / 4 ≡ − i mo d π 2 from P rop osition 3 . Hence x φ ( p 2 ) / 4 − i a nd x φ ( p 2 ) / 4 + i a re factored as ( x − g ) Q ( x ) mo d π 2 for x = g and ( x + g ) R ( x ) mo d π 2 for x = − g , resp ec- tively , where Q ( x ) and R ( x ) a re the p olyno mials in the indetermina te X with co efficient s in G π 2 . Moreov er, x φ ( p 2 ) / 2 + 1 can b e factored as ( x − g )( x + g ) A ( x ) mo d π 2 , where A ( x ) is the p o lynomials in the indeterminate X with coefficients in G π 2 .F urther more a ll comp onent s of a ny ro w of generator matrix do n ot con- sist o f zer o divisors since the g enerator p olynomial would b e selected a s a monic po lynomial. W e now ex plain how to construct cyclic co des ov er the other finite rings. Denote Z ∗ n by the set of multiplicativ e in verse elemen ts of Z n . If k ≥ 1 and k | n then the set Z ∗ n ( k ) is a subgroup of Z ∗ n , where Z ∗ n ( k ) =  x ∈ Z ∗ n : x ≡ 1 mo d k  . If s and t are relativ ely prime n umbers, then Z ∗ st ( s ) ∼ = Z ∗ t , Z ∗ st ( t ) ∼ = Z ∗ s . Prop osition 4 L et p 1 and p 2 b e o dd primes and let π 1 = a + bi and π 2 = c + di b e prime Gaussian inte gers, wher e p 1 6 = p 2 and p 1 = a 2 + b 2 = 4 n 1 + 1 and p 2 = c 2 + d 2 = 4 n 2 + 1 ( a, b, c, d, n 1 , n 2 ∈ Z ). If π 1 and π 2 ar e Gaussian inte gers, ther e ex ist any elements e , f of G ∗ π 1 π 2 satisfying e φ ( p 2 ) ≡ 1 mod π 1 π 2 and f φ ( p 1 ) ≡ 1 mo d π 1 π 2 . Pro of. Let p 1 and p 2 be distinct o dd primes. Then p 1 and p 2 are r elatively primes. Since s a nd t ar e r elatively prime n umbers, p 1 and p 2 can b e sele c ted as s and t , respectively , that is, s = a 2 + b 2 = 4 n 1 + 1 and t = c 2 + d 2 = 4 n 2 + 1. Using (1), we have Z s ∼ = G π 1 and Z t ∼ = G π 2 . It is clear that Z s.t ∼ = G π 1 .π 2 from Theorem 1 . Th us , we hav e G ∗ π 1 .π 2 ( π 1 ) ∼ = Z ∗ s.t ( s ) ∼ = Z ∗ t ∼ = G ∗ π 2 . G ∗ π 2 is a cyclic group b ecause π 2 is a prime Gaussia n integer. So, G ∗ π 1 .π 2 ( π 1 ) ha s a genera to r. Let’s call this generator e . Then e φ ( p 2 ) ≡ 1 mo d π 1 .π 2 . In the similar way , G ∗ π 1 .π 2 ( π 2 ) has a genera tor, let’s call f . Then f φ ( p 1 ) ≡ 1 mo d π 1 .π 2 since G ∗ π 1 .π 2 ( π 2 ) is isomorphic to G ∗ π 1 . Prop osition 5 L et p k b e k distinct prime inte ger, wher e p k is prime o dd inte- ger, π k = a k + ib k and p k = a 2 k + b 2 k = 4 n k + 1 for k = 1 , 2 , ..., m . . So, t her e exists an element e k of G ∗ π 1 .π 2 ...π m such t hat e φ ( p k ) k ≡ 1 mod π 1 .π 2 ...π m . Pro of. This is immediate from P r op osition 4. Theorem 3 L et p 1 and p 2 b e distinct prime o dd inte gers in Z and let π 1 = a + bi and π 2 = c + di b e p rime Gaussian inte gers in G , wher e p 1 = a 2 + b 2 = 4 n 1 + 1 , p 2 = c 2 + d 2 = 4 n 2 + 1 , n 1 , n 2 ∈ Z . Then t her e exists a cyclic c o de of length φ ( p 1 ) and φ ( p 2 ) over the ring G π 1 .π 2 . The gener ator p olynomial of this cyclic c o de is a first de gr e e monic p olynomial. 3 Pro of. F rom Prop osition 4, x φ ( p 2 ) − 1 can be fac to red as ( x − e ) .D ( x ) mo d π 1 π 2 since e φ ( p 2 ) ≡ 1 mo d π 1 .π 2 . If we take the generato r p olynomial a s g ( x ) = x − e , then the generator p olynomial g ( x ) generates the ge ner ator matrix, who se any r ow of all compo nen ts do not consist of zero divis ors. T o illustra te the constructio n of cyclic co des ov er so me finite rings, w e con- sider examples as follows. Example 1 The p olynomial x 10 + 1 factors over the ring G 3+4 i as ( x − 2) . ( x − 1 + i ) A ( x ) , wher e A ( x ) is a p olynomial in G 3+4 i [ X ] . If the gener ator p olynomial g ( x ) is taken as x 2 + (1 − 2 i ) x + ( − 2 + i ) , then the gener ator matrix and the p arity che ck matrix ar e as fol lows, r esp e ctively. G =             − 2 + i 1 − 2 i 1 0 0 0 0 0 0 0 0 − 2 + i 1 − 2 i 1 0 0 0 0 0 0 0 0 − 2 + i 1 − 2 i 1 0 0 0 0 0 0 0 0 − 2 + i 1 − 2 i 1 0 0 0 0 0 0 0 0 − 2 + i 1 − 2 i 1 0 0 0 0 0 0 0 0 − 2 + i 1 − 2 i 1 0 0 0 0 0 0 0 0 − 2 + i 1 − 2 i 1 0 0 0 0 0 0 0 0 − 2 + i 1 − 2 i 1             , H =  1 − (1 − 2 i ) − ( − 2 + i ) − (2 + i ) − (1 + i ) − (2 + i ) 3 i − 1 + i 2 i 0 0 1 − (1 − 2 i ) − ( − 2 + i ) − (2 + i ) − (1 + i ) − (2 + i ) 3 i − 1 + i 2 i  . L et us assume that at the r e c eiving end we get the ve ctor r =  − 2 + i 1 − 2 i 1 i 0 0 0 0 0 0  . First we c ompute the syndr ome S as fol lows: S = ix 3 + x 2 + (1 − 2 i ) x + ( − 2 + i ) x 2 + (1 − 2 i ) x + ( − 2 + i ) = (1 + 2 i ) x + (2 + i ) . Ther efor e, fr om T ab le I, it is se en that the syndr ome S ≡ i x 3 . Notic e that first we c ompute the syndr ome of the r e c eive d ve ctor to b e de c o de d. If this syndr ome disapp e ars in T able I, then its asso ciates che ck. Thus, the r e c eive d ve ct or r is de c o de d as c ( x ) = r ( x ) − ix 3 = x 2 + (1 − 2 i ) x + ( − 2 + i ) . Final ly we get c =  − 2 + i 1 − 2 i 1 0 0 0 0 0 0 0  . Example 2 L et p 1 = 5 , p 2 = 1 3 . L et the gener ator p olynomia l g ( x ) = x − 3 − i and the p arity che ck p olynomial h ( x ) = [ x 3 + (3 + i ) x 2 + (4 − i ) x + 2 − 2 i ] b e. Then we obtain the gener ator matrix G and p arity che ck matrix H as fol lows, r esp e ct ively. G =   − 3 − i 1 0 0 0 − 3 − i 1 0 0 0 − 3 − i 1   , H =  1 3 + i 4 − i 2 − 2 i  . Assume t hat r e c eive d ve ctor is r =  − 3 − i 1 i 0  . We c ompute the syn- dr ome as r ( x ) g ( x ) = ix 2 + x − 3 − i x − 3 − i = ( x − 3 − i )( ix + 3 i ) + (1 + 4 i ) . 4 Sinc e 1 + 4 i ≡ x 2 .i , the ve ctor r ( x ) is de c o de d as c ( x ) = r ( x ) − ix 2 = x − 3 − i . In T able II, c oset le aders and its syndr omes ar e given. T able 1: The c oset leader s and its syndromes. 0 0 x 6 ( − 2 − i ) x + 3 i 1 1 x 7 3 ix + (2 + i ) x x x 8 ( − 1 + 2 i ) x + 2 i x 2 ( − 1 + 2 i ) x + (2 − i ) x 9 2 ix + (1 − 2 i ) x 3 (2 − i ) x + (1 − 2 i ) x 1 0 − 1 x 4 ( − 2 − i ) x + ( − 1 − i ) x 1 1 x 11 = x 10 x x 5 ( − 1 − i ) x + (2 + i ) x 1 2 x 12 = x 10 x 2 T able 2: The c oset leaders and its syndromes. 0 0 1 1 (and its assoc iates) x 3 + i (and its asso ciates) x 2 4 − i (and its asso ciates) x 3 2 − 2 i (and its asso ciates) References [1] Hub er K., ”Co des Over Gaussian In tegers” IEEE T rans. Inform.Theory , vol. 40, pp. 207-21 6, jan. 1994 . [2] Hub er K., ”Co des Over Eisenstein-Ja cobi In tegers,” AMS, Co n temp. Math., vol. 1 58, pp. 165-179, 1994. [3] Hub er K., ”The MacWilliams theore m for tw o- dimensional mo dulo metrics” AAECC Springer V erlag , v o l. 8, pp. 4 1-48, 1997. [4] Neto T.P . da N., ”Lattice Cons tella tions and Codes F rom Quadratic Number Fields” IEEE T r ans. Inform. Theory , v ol. 47, pp. 1514-1 5 27, May 2001. [5] F an Y. and Gao Y., ”Co des Over Algebraic Integer Ring s o f Cycloto mic Fields” IEEE T r ans. Inform. Theory , v ol. 50, No. 1 jan. 2004. [6] Dr esden G. and Dymacek W.M., ”Finding F actors of F actor Rings Over The Gaussian Integers” The Mathematical Asso ciation of America, Monthly Aug-Sep. 2005 . 5