Cyclic Codes over Some Finite Quaternion Integer Rings

Cyclic Co des o v er Some F inite Quaternion Integer R ings Mehmet ¨ Ozen, Murat G ¨ uzeltepe Department of Mathematics, Sak ary a Un iversit y , TR54187 Sak arya , T urkey Abstract In this paper, cyclic co d es are established ov er some finite q uaternion integ er rings with respect to the quaternion Mannheim distance, and de- codin g algorithm for these co des is given. 2000 AMS Classification: 94 B05, 94B1 5, 94B35, 9 4B60 Keywords: B lock codes, Mannheim distance, Cyclic co des, Syndrome deco ding 1 In tro duction Mannheim distance, whic h is m uch better suited for co ding over t wo dimensional signal space than the Hamming distance, was intro duced by Hub er [1]. More- ov er , Huber c o nstructed one Mannheim erro r correcting co des , which are s uit- able for quadr ature amplitude mo dulation (QAM)-type mo dula tio ns [1]. Cyclic co des over some finite rings with resp ect to the Mannheim metric were obtained by using Gaus s ian in tegers in [2]. Later, in [3], using quaternio n Mannheim met- ric, also ca lled Lipschitz metric [4], p erfect codes o ver some finite qua ternion int eger rings were o btained and these codes were decoded. The re st of this pa per is o r ganized a s follows. In Section I I, qua ternion int egers and some fundamen tal algebra ic concepts hav e b een cons idered. In Section II I, we construct cyclic co des over some quater nion integer r ings with resp ect to quaternion Mannheim metric. 2 Quaternion In tegers Definition 1 The Hamilton Qu atern ion Alge br a over the set of the r e al nu m- b ers ( R ), d enote d by H ( R ) , is the asso ciative un ital algebr a give n by the fol low- ing r epr esentation: i) H ( R ) is the fr e e R mo dule over t he symb ols 1 , i, j, k , t hat is, H ( R ) = { a 0 + a 1 i + a 2 j + a 3 k : a 0 , a 1 , a 2 , a 3 ∈ R } ; ii)1 is the multiplic ative un it; iii) i 2 = j 2 = k 2 = − 1 ; iv) ij = − j i = k , i k = − k i = j, j k = − k j = i [5 ]. The set H ( Z ), H ( Z ) = { a 0 + a 1 i + a 2 j + a 3 k : a 0 , a 1 , a 2 , a 3 ∈ Z } , is a subset of H ( R ), where Z is the set of all integers. If q = a 0 + a 1 i + a 2 j + a 3 k is 1 a qua ternion integer, its conjugate q ua ternion is q = a 0 − ( a 1 i + a 2 j + a 3 k ).The norm o f q is N ( q ) = q .q = a 2 0 + a 2 1 + a 2 2 + a 2 3 . A quaternio n integer consists of t wo parts whic h ar e the complete part and the vector part. Let q = a 0 + a 1 i + a 2 j + a 3 k b e a quaternion integer. Then its complete part is a 0 and its vector part is a 1 i + a 2 j + a 3 k . The co mmutative proper t y of multiplication do e s not hold for quater nion integers. H ow ever, if the vector pa rts o f qua ternion integers are parallel to eac h other, then their pro duct is commut ative. Define H ( K 1 ) a s follows: H ( K 1 ) = { a 0 + a 1 ( i + j + k ) : a 0 , a 1 ∈ Z } which is a subset of q uaternion integers. The commutativ e prop erty of mult i- plication holds ov e r H ( K 1 ). Theorem 1 F or every o dd, r ational prime p ∈ N , ther e exists a prime π ∈ H ( Z ) , su ch that N ( π ) = p = π π . In p articular, p is not prime in H ( Z ) [5]. Corollary 1 π ∈ H ( Z ) is prime in H ( Z ) if and only if N ( π ) is prime in Z [5]. Theorem 2 If a and b ar e r elatively prime int e gers then H ( K 1 ) / h a + b ( i + j + k ) i is isomorphic to Z a 2 +3 b 2 [3, 5, 7]. 3 Cyclic Co des o v er Quaternion In teger Rings Let H ( K 1 ) π k be the res idue class of H ( K 1 ) π mo dulo π k , where k is any p o sitive int eger and π is a prime qua ternion in teger. According to the mo dulo function µ : Z P k → H ( K 1 ) π k defined b y g → g − [ g . π π .π ] π ( mo d π k ) (1) H ( K 1 ) π k is isomorphic to Z p k , where p = π π and p is an odd prime. A quater- nion cyclic co des C of length n is a linear code C of length n with proper t y ( c 0 , c 1 , ..., c n − 1 ) ∈ C ⇒ ( c n − 1 , c 0 , c 1 , ..., c n − 2 ) ∈ C . In this case, w e have a bijective H ( K 1 ) n π k → H ( K 1 ) π k [ x ] / ( x n − 1) ( c 0 , c 1 , ..., c n − 1 ) 7→ c 0 + c 1 x + · · · + c n − 1 x n − 1 + ( x n − 1) (2) T o put it simply , we wr ite c 0 + c 1 x + · · · + c n − 1 x n − 1 for c 0 + c 1 x + · · · + c n − 1 x n − 1 + ( x n − 1). A nonempty set of H ( K 1 ) n π k is a H ( K 1 ) π k -cyclic co de if and only if its image under (2) is a n ideal of H ( K 1 ) π k [ x ] / ( x n − 1). More infor ma tion o n cyclic co des can be found in [6]. Definition 2 L et α, β ∈ H ( K 1 ) π and γ = β − α = a + b ( i + j + k ) (mo d π ) , wher e π is a prime quaternion inte ger. L et the quaternion Mannheim weight of γ b e define d as w QM ( γ ) = | a | + 3 | b | the quatern ion Mannheim distanc e d QM b etwe en α and β is define d as d QM ( α, β ) = w QM ( γ ) . [ 3 ] 2 Prop ositio n 1 L et π = a + b ( i + j + k ) b e a prime in t he s et H ( K 1 ) and let p = a 2 + 3 b 2 b e prime in Z . If g is a gener ator of H ( K 1 ) ∗ π 2 , then g φ ( p 2 ) / 2 ≡ − 1 ( mo d π 2 ) . Pro of. If N ( π ) is a prime integer in Z , then the complete par t a nd the co ef- ficient of the vector part o f π 2 are rela tively integer. So, Z p 2 is isomo rphic to H ( K 1 ) π 2 (See Theorem 2). If g is a gener ator of H ( K 1 ) ∗ π 2 , then g , g 2 , ..., g φ ( p 2 ) constitute a reduced res idue sys tem mo dulo π 2 in H ( K 1 ) π 2 . Therefor e, there is a p ositive integer k as g k ≡ − 1 (mo d π 2 ), where 1 ≤ k ≤ φ ( p 2 ). Hence, we can infer g 2 k ≡ 1 (mo d π 2 ). Since φ ( p 2 )   2 k a nd 2 ≤ 2 k ≤ 2 φ ( p 2 ), we obtain φ ( p 2 ) = k or φ ( p 2 ) = 2 k . If φ ( p 2 ) was equal to k , w e should hav e π 2   2, but th is would co ntradict the fact that N ( π 2 ) > 2. Prop ositio n 2 L et π k = a k + b k ( i + j + k ) b e distinct p rimes in H ( K 1 ) and le t p k = a 2 k + 3 b 2 k b e distinct primes in Z , wher e k = 1 , 2 , ..., m . If g is a gener ator of H ( K 1 ) ∗ π k , then g φ ( p k ) / 2 ≡ − 1 ( mo d π k ) . Pro of. This is certain from Prop ositio n 1 . Theorem 3 L et π = a + b ( i + j + k ) b e a prime in H ( K 1 ) and let p = a 2 + 3 b 2 b e a prime in Z , wher e a, b ∈ Z . Then, cyclic c o des whose lengths ar e φ ( p 2 ) / 2 ar e obtaine d. Pro of. H ( K 1 ) ∗ π 2 has a genera tor since Z p 2 ∼ = H ( K 1 ) π 2 . Let the genera tor b e g . Then we get g φ ( p 2 ) = 1 and g φ ( p 2 ) / 2 = − 1. Hence, we can write x φ ( p 2 ) / 2 + 1 = ( x − g ) Q ( x ) ( mo d π 2 ) (for x = g ) . In this situation, ( x − g ) is an ideal of H ( K 1 ) π 2 [ x ] .D x φ ( p 2 ) / 2 + 1 E , i.e., it gen- erates a cyclic code. If the ge nerator polynomial is tak en as a mo nic p olynomial, all compo nen ts of a ny row of the gene r ator matrix do not consist of z e r o divisors . Therefor e, these co des are free H ( K 1 ) π 2 mo dules. Prop ositio n 3 L et π 1 = a + b ( i + j + k ) , π 2 = c + d ( i + j + k ) b e primes in H ( K 1 ) and let p 1 = a 2 + 3 b 2 , p 2 = c 2 + 3 d 2 b e primes in Z . Then, t her e ar e two elements of H ( K 1 ) ∗ π 1 π 2 such t hat e φ ( p 2 ) ≡ 1 ( mo d π 1 π 2 ) and f φ ( p 1 ) ≡ 1 ( mo d π 1 π 2 ) . Pro of. Since p 1 and p 2 are relatively prime integers in Z , π 1 and π 2 are rela- tively primes in H ( K 1 ). Using the basic algebra ic knowledge and the function (1), we get Z p 1 ∼ = H ( K 1 ) π 1 , Z p 2 ∼ = H ( K 1 ) π 2 and Z p 1 p 2 ∼ = H ( K 1 ) π 1 π 2 . Mor eov e r, we obtain a s f ollows: H ( K 1 ) ∗ π 1 π 2 ( π 1 ) ∼ = Z ∗ p 1 p 2 ( p 1 ) ∼ = Z ∗ p 2 ∼ = H ( K 1 ) ∗ π 2 , H ( K 1 ) ∗ π 1 π 2 ( π 2 ) ∼ = Z ∗ p 1 p 2 ( p 2 ) ∼ = Z ∗ p 1 ∼ = H ( K 1 ) ∗ π 1 . Since π 2 is a prime quater nion integer, H ( K 1 ) ∗ π 2 is a cyclic gr oup. There fore, H ( K 1 ) ∗ π 2 has a genera tor. So, H ( K 1 ) ∗ π 1 π 2 ( π 1 ) has a genera to r, either. Let the generator b e e . Then e φ ( p 2 ) ≡ 1 (mo d π 1 π 2 ). In the sa me way , H ( K 1 ) ∗ π 1 π 2 ( π 2 ) has a g e nerator. Suppo se that f is the generator of H ( K 1 ) ∗ π 1 π 2 ( π 2 ). Then f φ ( p 1 ) ≡ 1 ( mo d π 1 π 2 ). 3 Prop ositio n 4 L et π k = a k + b k ( i + j + k ) b e a prime in H ( K 1 ) and let p k = a 2 k + 3 b 2 k b e distinct o dd primes in Z . Then, ther e is an element e k of H ( K 1 ) ∗ π 1 π 2 ...π k such t hat e φ ( p k ) k ≡ 1 ( mo d π 1 π 2 ...π k ) , k = 1 , 2 , ..., m . Pro of. This is clear from Prop osition 3. Theorem 4 L et π 1 = a + b ( i + j + k ) , π 2 = c + d ( i + j + k ) b e primes in H ( K 1 ) and let p 1 = a 2 + 3 b 2 , p 2 = c 2 + 3 d 2 b e o dd primes in Z . Then, we c an always write cyclic c o des of length φ ( p 1 ) and φ ( p 2 ) over H ( K 1 ) π 1 π 2 .Mor e over, the gener ator p olynomials of t hese c o des ar e first de gr e e monic p olynomials. Ther efor e, these c o des ar e fr e e H ( K 1 ) π 1 π 2 mo dule. Pro of. F rom Pr op osition 3, we can find an element of H ( K 1 ) π 1 π 2 such tha t e φ ( p 2 ) ≡ 1 ( mod π 1 π 2 ). Thu s, we fa ctorize the p olynomial x φ ( p 2 ) − 1 over H ( K 1 ) π 1 π 2 as x φ ( p 2 ) − 1 = ( x − e ) D ( x )(mod π 1 π 2 ). If we take the gener ator po lynomial a s g ( x ) = x − e , then the g enerator p olynomial g ( x ) forms the gen- erator matrix who se all comp onents of an y rows do not cons is t o f zer o divisors. W e now co ns ider a simple example with regard to Theo rem 3. Example 1 L et π b e 2 + i + j + k . The p olynomial x 21 + 1 factors over H ( K 1 ) π 2 as x 21 + 1 = ( x − α ) . ( x 20 + αx 19 + α 2 x 18 + α 3 x 17 + ... + α 19 x + α 20 ) , wher e α = 1 − i − j − k . The p owers of α ar e shown in T able I. If we cho ose t he gener ator p olynomial as g ( x ) = x − α , then the gener ator matrix is as fol lows: G =         − α 1 0 0 · · · 0 0 − α 1 0 · · · 0 0 0 − α 1 . . . 0 . . . . . . . . . . . . 0 0 · · · 0 − α 1         20x21 . The c o de C gener ate d by the gener ator matrix G c an c orr e ct one err or having quaternion Mannheim weight of one. T able I: Pow ers of t he ele ment α = 1 − i − j − k which is root of x 3 + 1. s α s s α s s α s s α s 0 1 6 3 + i + j + k 12 4 − 2 i − 2 j − 2 k 18 − 6 1 1 − i − j − k 7 − 6 − i − j − k 13 2 i + 2 j + 2 k 19 5 + i + j + k 2 − 1 + 2 i + 2 j + 2 k 8 2 14 5 − 2 i − 2 j − 2 k 20 − 3 + i + j + k 3 4 − i − j − k 9 2 − 2 i − 2 j − 2 k 1 5 1 + i + j + k 21 − 1 4 2 − i − j − k 10 − 3 16 4 22 − α = − 1 + i + j + k 5 i + j + k 11 − 4 − i − j − k 17 5 23 − α 2 = 1 − 2 i − 2 j − 2 k References [1] Huber K., ”Co des Over Gaussian Integers” IEE E T rans . Infor m.Theory , vol. 40, pp. 207-21 6, Jan. 1994 . [2] M. ¨ Ozen and M. G ¨ uzeltepe , ”Cy clic Co des ov er Some Finite Rings ” (Sub- mitted 2009 ). 4 [3] M. ¨ Ozen a nd M. G ¨ uzeltep e, ” Co des ov er Quaternion Integers” (Submitted 2009). [4] C. Mar tinez, E. Staffor d, R. Beiv ide , E. Gabidulin, ”Perfect Co des ov er Lipschitz In teger s” IE EE In t. Sympos ium, ISIT 2 007. [5] G. Davidoff, P . Sarnak, A. V alette, ”E lemen tary Num b er Theor y , Gro up Theory , Raman ujan Graphs”, Cam br idge Univ ers it y Pres, 2003. [6] F. J. Macwilliams and N. J. SLOANE, ”The Theory of Err or Corre c ting Co des”, North Holland Pub. Co., 1977 . [7] G. Dres den and W. M. Dymacek, ”Finding F actor s o f F actor Rings over the Ga ussian Integers” The Ma thematical Asso cia tion of America, Monthly Aug-Sep. 2005 . [8] I. Ziven, H.S. Zuck erman and H.L. Montgomery , ”An Introduction to the Num b er Theory” John Wiley and Sons, Inc., 1991. 5