Cocyclic Butson Hadamard matrices and Codes over Z n via the Trace Map

Transcription

1 Contemporary Mathematcs Cocyclc Butson Hadamard matrces and Codes over Z n va the Trace Map N. Pnnawala and A. Rao Abstract. Over the past couple of years trace maps over Galos felds and Galos rngs have been used very successfully to construct cocyclc Hadamard, complex Hadamard and Butson Hadamard matrces and subsequently to generate smplex codes over Z 4, Z s and Z p and new lnear codes over Z p s. Here we defne a new map, the trace-le map and more generally the weghted-trace map and extend these technques to construct cocyclc Butson Hadamard matrces of order n m for all n and m and lnear and non-lnear codes over Z n.. Introducton The cocyclc map has been used to construct Hadamard matrces (see []) and these Hadamard matrces were found to yeld bnary extremal self-dual codes []. The nature of the cocyclc map allowed for substantal cut-down n the computatonal tme needed to generate the matrces and then the codes. In [] the authors exploted ths property to construct cocyclc Complex and Butson Hadamard matrces by defnng the cocycle maps va the trace maps over Galos rngs GR(4, m) and GR( e, m) respectvely. In [3], ths method was extended to construct some new lnear codes over Z p e for prme p > and postve nteger e. A challengng open problem was the extenson of ths method to construct Butson Hadamard matrces of order n for any postve nteger n. The prme factorzaton of n,.e., n = p e pe... pe and the somorphsm Z n = Z e p Z e p... Z e p paves the way to focus our attenton on the rng R(n, m) = GR(p e, m) GR(pe, m)... GR(pe, m), where m s a postve nteger. However there s no nown map over ths rng smlar to the trace map over Galos rngs and Galos felds. In ths paper, we defne a new map, the trace-le map, over the rng R(n, m). A generalzaton of ths map, called the weghted-trace map, s used n [9] for Fourer transforms. These maps satsfy fundamental propertes parallel to the other trace maps, and can be used n a smlar manner to the trace maps n [] and [3] to frst unformly construct 000 Mathematcs Subject Classfcaton. Prmary 94B05; Secondary T7. Key words and phrases. Cocycle, complex Hadamard, Butson, smplex codes, Trace, exponent. The wor of N. Pnnawala was carred out durng hs Ph.D. studes and was supported by a School of Mathematcal and Geospatal Scences Scholarshp. c 008 Amercan Mathematcal Socety

2 N. PINNAWALA AND A. RAO cocyclc Butson Hadamard matrces of any order n and then lnear and non-lnear codes over Z n. A lnear code C of length n over the ntegers modulo (.e., Z = {0,,,..., }) s an addtve subgroup of Z n. An element of C s called a codeword and a generator matrx of C s a matrx whose rows generate C. The Hammng weght W H (x) of an n-tuple x = (x, x,..., x n ) n Z n s the number of nonzero components of x and the Lee weght W L (x) of x s n = mn {x, x }. The Eucldean weght W E (x) of x s n = mn {x, ( x ) } and the Chnese Eucldean weght W CH (x) of x s n { ( cos πx )}. The Hammng, Lee, Eucldean = and Chnese Eucldean dstances between x, y Z n d H (x, y) = W H (x y), d L (x, y) = W L (x y), d E (x, y) = W E (x y) and d CE (x, y) = W CE (x y) respectvely. A cocycle s a set mappng, ϕ : G G C, whch satsfes are defned and denoted as ϕ(a, b)ϕ(ab, c) = ϕ(b, c)ϕ(a, bc), a, b, c G, where G s a fnte group and C s a fnte abelan group. The matrx M ϕ = [ϕ(x, y)] x,y G s called a cocyclc matrx. Butson Hadamard matrces were frst ntroduced by Butson n 96 [4]. A square matrx H of order n all of whose elements are complex p th roots of unty (p not necessarly a prme) s called a Butson Hadamard matrx, denoted by BH(n, p), ff HH = ni, where H s the conjugate transpose of H and I s the dentty matrx of order n. In 979, Drae [6] ntroduced generalzed Hadamard matrces. A square matrx H = [h j ] of order n over a group G s called a generalzed Hadamard matrx GH(n, G) f for j the sequence {h x h jx } wth x n contans every element of G equally often. For prme p the defnton of a BH(n, p) and a GH(n, C p ) are equvalent, where C p denotes the multplcatve group of all complex p th roots of unty. On the other hand, f p = mt, where m s a prme and t >, then there exsts a Butson Hadamard matrx of order m over C p, but certanly no generalzed Hadamard matrx of order m over C p (Remar.3, [6]). The authors have been unable to fnd a reference for unform constructon of Butson Hadamard matrces. Ths paper provdes such a unform constructon. In Secton we study the Galos rng GR(p e, m) and the propertes of the trace map over GR(p e, m). A cocycle over GR(p e, m) s defned and the cocyclc Butson Hadamard matrx of order p em s constructed. Ths matrx s then used to construct lnear codes over Z p e. Secton 3 detals the rng R(n, m) = GR(p e, m) GR(p e, m), n = pe pe, and the propertes of the trace-le map over R(n, m). The trace-le map s then used to construct cocyclc Butson Hadamard matrces of order n m and the exponent matrces are used to construct cocyclc codes over Z n. In addton, these results are easly extended to construct codes over Z n for n = p e pe... pe. We also pont out the relatonshp to the senary smplex codes of type α n [8]. In Secton 4 the Hammng, Lee, Eucldean and Chnese Eucldean dstances of these codes are calculated. A further generalzaton of the trace-le map, called the weghted-trace map (whch frst appeared n [9]) s studed n Secton 5 and used to construct cocyclc Butson Hadamard matrces and consequently to construct non-lnear codes over Z n. Fnally, n Secton 6, we summarze the results of ths paper.

3 COCYCLIC BUTSON HADAMARD MATRICES AND CODES VIA THE TRACE MAP 3. The Galos rng GR(p e, m), the trace map and cocyclc Z p e - lnear codes For the study of Z p e-codes, we frst need a bref revew of the Galos rng of characterstc p e and dmenson m. For more detals on Galos rngs of ths type, the reader s referred to [] and [4]. Here we gve n detal the results for prmes p >, but the case p = s smlar and detals can be found n []. Let p > be a prme and e be a postve nteger. The rng of ntegers modulo p e s the set Z p e = {0,,,..., p e }. Let h(x) Z p e[x] be a basc monc rreducble polynomal of degree m that dvdes x pm. The Galos rng of characterstc p e and dmenson m s defned as the quotent rng Z p e[x]/(h(x)) and s denoted by GR(p e, m). The element ζ = x + (h(x)) s a root of h(x) and consequently ζ s a prmtve (p m ) th root of unty. Therefore we say that ζ s a prmtve element of GR(p e, m) and GR(p e, m) = Z p e[ζ]. Hence GR(p e, m) =<, ζ, ζ,..., ζ m > and GR(p e, m) = p em. It s well nown that each element u GR(p e, m) has a unque representaton: u = e =0 p u, where u T = {0,, ζ, ζ,..., ζ pm }. Ths representaton s called the p-adc representaton of elements of GR(p e, m) and the set T s called the Techmüller set. Note that u s nvertble f and only f u 0 0. Thus every non-nvertble element of GR(p e, m) can be wrtten as u = e = p u, =,,..., e, and we can represent all the elements of GR(p e, m) n the form u () = e = p u, = 0,,,..., e. Usng the p- adc representaton of the elements of GR(p e, m), the Frobenus automorphsm f s defned n [3], [5] and [4] as f : GR(p e, m) GR(p e, m) f(u) = e =0 p u p. Note that when e =, f s the usual Frobenus automorphsm for the Galos feld GF (p, m) (see [0]). The trace map over GR(p e, m) s then defned by T r : GR(p e, m) Z p e T r(u) = u + f(u) + f (u) f m (u). From the defnton of f and T r the trace map satsfes the followng propertes: For any u, v GR(p e, m) and α Z p e. T r(u + v) = T r(u) + T r(v).. T r(αu) = αt r(u).. T r s surjectve. In addton to these propertes the trace map also satsfes the followng property. Theorem.. [[3], Lemma.] Gven a Galos Rng GR(p e, m), let D = {p t t = 0,,..., p e } Z p e and u () be an element n GR(p e, m), as defned above. As x ranges over GR(p e, m), T r(xu () ) maps to each element n D equally often,.e., p e(m )+ tmes, where = 0,,,..., e. We are now n a poston to use the trace map to construct Butson Hadamard matrces and lnear codes over Z p e. Let ω = exp( π ) be the complex th root of unty and C be the multplcatve group of all complex th roots of unty..e., C = {, ω, ω,..., ω }. It s well nown that (.) S = ω j = 0. j=0

4 4 N. PINNAWALA AND A. RAO Let H = [h,j ] be a square matrx over C. The matrx E = [e,j ], e,j Z, whch s obtaned from H = [ω e,j ] = [h,j ] s called the exponent matrx assocated wth H. Theorem.. [[3], Proposton 3.] Let p be a prme, p >. Let GR(p e, m) be the Galos rng of characterstc p e and C p e be the multplcatve group of all complex p e th roots of unty.. The set mappng ϕ : GR(p e, m) GR(p e, m) C p e ϕ(c, c j ) = (ω) T r(ccj) s a cocycle.. The matrx M ϕ = [ϕ(c, c j )] c,c j GR(p e,m) s a Butson Hadamard matrx of order p em.. The rows of the exponent matrx of M ϕ (.e., A = [T r(c c j )] c,c j GR(p e,m)) form a lnear code over Z p e ( wth )] parameters [n,, d L ] = [p em, m, p e(m ) p e p (e ) Rng R(n, m) and cocyclc Z n -lnear codes Let R(n, m) be the drect product of Galos rngs. In ths secton we wll loo at the structure of the rng R(n, m) and defne a new map over R(n, m) usng the trace maps over the component Galos Rngs. We call ths map the trace-le map snce t satsfes propertes smlar to that of the trace maps over Galos rngs and Galos felds. We then use ths map to construct cocyclc Butson Hadamard matrces of order n m for all postve ntegers n and m. In the frst nstance let us loo at the case n = p e pe, where p p are prmes and e, e are postve ntegers. It s well nown that Z n = Zp e Z e p and hence for any postve nteger m, Z m n = (Z e p Z e p ) m. For more detals on these results see for example [7]. Let f (x) and f (x) be basc monc rreducble polynomals of degree m over Z e p and Z e p respectvely. As n Secton the Galos rngs of characterstcs p e and p e and common dmenson m are defned as the quotent rngs Z e p [x]/(f (x)) and Z e p [x]/(f (x)) respectvely. These rngs are denoted by GR(p e, m) and GR(pe, m). If ζ and ζ are defned to be ζ = x + (f (x)) and ζ = x+(f (x)), the two rngs can then be expressed as GR(p e, m) =<, ζ, ζ,..., ζ m > and GR(p e, m) =<, ζ, ζ,..., ζ m > respectvely. Ths tells us that GR(p e, m) = Z p e [ζ ] and GR(p e, m) = Z p e [ζ ]. Hence any element a GR(p e, m) can be expressed as an m-tuple a = (a 0, a,..., a m ) over Z e p whle b GR(p e, m) as b = (b 0, b,..., b m ) over Z e p. Now consder the drect product of the two Galos rngs. Let R(n, m) =, m) GR(pe, m). Any element c R(n, m) can be wrtten as c = (a, b), where a GR(p e, m) and b GR(pe, m) and further as c = (a 0, a,... a m, b 0, b,..., b m ). Snce Z n = Zp e Z e p, c can also be wrtten as an m-tuple c = (c 0, c,..., c m ) over Z n, where c = (a, b ) = 0,,,..., m, a Z e p GR(p e and b Z e p. Let c, c be elements n R(n, m). It s easy to see that R(n, m) s a rng under the addton c + c = ((c 0 + c 0), (c + c ),..., (c m + c m ) and the multplcaton cc = (c 0 c 0, c c,..., c m c m ). GR(p e, m) GR(pe, m). Also R(n, m) = n m = (p e pe )m =

5 COCYCLIC BUTSON HADAMARD MATRICES AND CODES VIA THE TRACE MAP 5 (3.) In ths context, t s well nown that: f p s a prme and a s any nteger then a p a (mod p). The next result follows mmedately from the Chnese remander theorem. Lemma 3.. Let n = p e pe. Then Z n = Z p e Z p e there exst α Z e p and α Z e p such that α = (α p e Z n = {(α, α ) α Z e p, α Z e p }. and gven α Z n + α p e ) mod n. Thus Theorem 3. (Trace-le map). Let T r and T r be the trace maps over GR(p e, m) and GR(pe, m) respectvely. For any c = (c, c ) R(n, m), the map T over R(n, m) defned by T : R(n, m) Z n T (c) = p e T r (c ) + p e T r (c ) satsfes the followng propertes: For any c, c R(n, m) and α Z n. T (c + c ) = T (c) + T (c ).. T (αc) = αt (c).. T s surjectve. Proof:. Let c, c R(n, m) = GR(p e, m) GR(pe, m). Then c = (c, c ) and c = (c, c ), where c, c GR(p e, m) and c, c GR(p e, m). Snce c + c = (c + c, c + c ) we have T (c + c ) = p e T r (c + c ) + p e T r (c + c ) = (p e T r (c ) + p e T r (c )) + (p e T r (c ) + p e T r (c )) = T (c) + T (c ).. Let any α Z n and c R(n, m). T (αc) = p e T r (αc ) + p e T r (αc ) = p e (αc + α p g (c ) α pm g (c )) +p e (αc + α p g (c ) α pm g (c )) = p e α(t r (c )) + p e α(t r (c )) from (3.) = αt (c). Here g and g are the Frobenus automorphsms over GR(p e, m) and GR(p e, m) respectvely.. Snce T r and T r are both surjectve and not dentcally zero, there exst elements c GR(p e, m) and c GR(p e, m) such that T r (c ) = and T r (c ) =. Then c = (c, c ) R(n, m) and T (c) = p e T r (c ) + p e T r (c ) = p e + pe. For all α Z n we have proved n () that T (αc) = αt (c) and snce p e + pe s not a multple of ether p or p, T (αc) = αt (c) should represent every element n Z n and hence T s surjectve. Snce the trace-le map s a combnaton of the Galos rng traces, t s equdstrbuted, just as the component trace maps are equ-dstrbuted. We prove ths n the next theorem.

6 6 N. PINNAWALA AND A. RAO Theorem 3.3. For any c R(n, m), as x ranges over R(n, m), T (cx) taes each element n { } (3.) S,j = p p j n t t = 0,,,..., p pj equally often p p j nm tmes, where 0 e and 0 j e. Proof: We frst prove that T (cx) S,j. Snce c, x R(n, m), c = (c, c ) and x = (x, x ), where c, x GR(p e, m) and c, x GR(p e, m). In the case c = 0, T (cx) = 0. If c 0 and both c and c are non-zero, then as they are both elements of Galos Rngs, ther p-adc representatons are gven by: e c = u () = p u ; 0 e, u 0. = e c = u (j) = p u ; 0 j e, u j 0. =j Here u and u are n the Techmüller sets of the respectve Galos rngs. From Theorem., as x ranges over R(n, m), snce T (cx) = p e T r (c x )+p e T r (c x ), the two trace maps T r (c x ) and T r (c x ) wll tae values n the sets D = {p t 0 t p e } and D j = {p j t 0 t p e j } respectvely. Thus T (cx) {p e p t + p e pj t 0 t p e, 0 t p e j } = {p p j (pe j t + p e t ) 0 t p e, 0 t p e j }. Snce the calculaton are done modulo n, {(p e j t + p e t ) 0 t t ) 0. Hence T (cx) {p p j t 0 t p e, 0 t p e j } Z n. From Lemma 3., {(p e j t + p e t p e, 0 t p e j } = Z p e p e j p e p e j } = S,j. If c 0 and c = 0 (or c = 0) then T (cx) = p e T r (c x ) (respectvely T (cx) = p e T r (c x )), and we are reduced to the Galos rng case. From Theorem., T r (c x ) D j (respectvely T r (c x ) D ) whch mples T (cx) {p e pj t 0 s p e j } = S 0,j (respectvely T (cx) S,0 ). In addton T r (c x ) ( respectvely T r (c x )) taes each value n D (respectvely D j ) equally often p e(m )+ (respectvely p e(m )+j ). Hence T (cx) wll tae each value n S,j, equally often p e(m )+ p e(m )+j = p p j nm tmes. Snce the map T satsfes propertes smlar to those satsfed by the trace map over Galos felds and Galos rngs, we call t the trace-le map. Example 3.4. Consder the rng R(6, ) = GF (, ) GF (3, ) and the rreducble polynomals f(x) = x + x + over Z and g(x) = x + x + over Z 3. Thus GF (, ) = Z [x]/(f(x)) and GF (3, ) = Z 3 [x]/(g(x)). If ζ = (f(x)) + x then f(ζ ) = 0 and hence GF (, ) = Z [ζ ]. Smlarly f ζ = (g(x)) + x then g(ζ ) = 0 and hence GF (3, ) = Z 3 [ζ ].

7 COCYCLIC BUTSON HADAMARD MATRICES AND CODES VIA THE TRACE MAP 7 by The Frobenus automorphsms f and f over GF (, ) and GF (3, ) are gven f : GF (, ) GF (, ) and f : GF (3, ) GF (3, ) f (c ) = c f (c ) = c 3 respectvely. The trace maps T r and T r over GF (, ) and GF (3, ) are gven by T r : GF (, ) Z and T r : GF (3, ) Z 3 T r (c ) = c + f (c ) T r (c ) = c + f (c ) respectvely. Table llustrates the values of the trace maps. Element c T r (c ) 00 = = + 0 Element c T r (c ) 0 = 0 + ζ 00 = ζ = + ζ 0 = ζ 0 = + ζ 0 = 0 + ζ ζ ζ 3 = + ζ ζ 0 = + 0 ζ 0 = 0 + ζ ζ 5 = + ζ ζ 6 0 = + ζ ζ 7 Table. Trace map values over GF (, ) (left) and GF (3, ) (rght) The trace-le map T over the rng R(6, ) s defned as follows: T : R(6, ) Z 6 ; T (c) = 3T r (c ) + T r (c ), where c GF (, ) and c GF (3, ). Snce Z 6 = Z Z 3, elements of Z 6 can be represented by 0=(0,0), =(,), =(0,), 3=(,0), 4=(0,), 5=(,). Table llustrates the elements of R(6, ) and the values of the trace-le map over R(6, ). We are now n a poston to defne a cocycle usng the trace-le map. Theorem 3.5. Let ω = exp ( ) π n be a complex n th root of unty, where n = p e pe and C n be the set of all complex n th roots of unty.. The set mappng ϕ : R(n, m) R(n, m) C n ; ϕ(a, b) = ω T (ab) s a cocycle.. The matrx M ϕ = [ϕ(a, b)] a,b R(n,m) s a Butson Hadamard matrx of order n m.. The rows of the exponent matrx assocated wth M ϕ, (.e., A = [T (ab)] for a, b R(n, m)), form a lnear code over Z n wth parameters [n, ] = [n m, m]. In the case p < p and e e, the mnmum Hammng weght s gven by d H = (n p e pe )n m. Proof:

8 8 N. PINNAWALA AND A. RAO c c = (c, c ) T (c) c c = (c, c ) T (c) 00 (00)(00) = ((00), (00)) 0 0 ()(00) = ((0), (0)) 0 (00)() = ((0), (0)) 5 ()() = ((), ()) 0 (00)(0) = ((00), (0)) 4 ()(0) = ((0), ()) 0 03 (00)(0) = ((0), (00)) 3 3 ()(0) = ((), (0)) 5 04 (00)(0) = ((00), (0)) 4 ()(0) = ((0), ()) 4 05 (00)() = ((0), (0)) 5 ()() = ((), ()) 3 0 (0)(00) = ((00), (0)) 4 30 (0)(00) = ((0), (00)) 0 (0)() = ((0), ()) 3 3 (0)() = ((), (0)) 5 (0)(0) = ((00), ()) 3 (0)(0) = ((0), (0)) 4 3 (0)(0) = ((0), (0)) 33 (0)(0) = ((), (00)) 3 4 (0)(0) = ((00), ()) 0 34 (0)(0) = ((0), (0)) 5 (0)() = ((0), ()) 5 35 (0)() = ((), (0)) 40 (0)(00) = ((00), (0)) 50 ()(00) = ((0), (0)) 4 4 (0)() = ((0), ()) 5 ()() = ((), ()) 3 4 (0)(0) = ((00), ()) 0 5 ()(0) = ((0), ()) 43 (0)(0) = ((0), (0)) 5 53 ()(0) = ((), (0)) 44 (0)(0) = ((00), ()) 4 54 ()(0) = ((0), ()) 0 45 (0)() = ((0), ()) 3 55 ()() = ((), ()) 5 Table. Trace-le map values over R(6,). Let any a, b, c R(n, m). Then ϕ(a, b) = ω T (ab) ϕ(a + b, c) = ω T ((a+b)c) T (ac)+t (bc) = ω ϕ(b, c) = ω T (bc) ϕ(a, b + c) = ω T (a(b+c)) T (ab)+t (ac) = ω From these equatons we have ϕ(a, b)ϕ(a + b, c) = ϕ(b, c)ϕ(a, b + c) Thus ϕ s a cocycle. Ths proof also follows from Proposton.4 [].. Let M ϕ = [ϕ(a, b)] a,b R(n,m). To prove that M ϕ M ϕ = n m I, consder the sum (3.3) S = ϕ(a, x)ϕ(x, b), x R(n,m) where ϕ(x, b) s the complex conjugate of ϕ(x, b). From the propertes of the trace-le map (Theorem 3.) (3.4) S = x R(n,m) ω T (x(a b)). When a = b, S = n m. When a b, from Theorem 3.3 we have

9 COCYCLIC BUTSON HADAMARD MATRICES AND CODES VIA THE TRACE MAP 9 (3.5) (3.6) S = p p j nm n p pj t=0 ω p pj t, where 0 e and 0 j e. From the equaton (.) we have S = 0. Thus the matrx M ϕ s a Butson Hadamard matrx of order n m. Let B = [T r (c α c α )] for c α, c α GR(p e, m) and D = [T r (c β c β )] for c β, c β GR(p e, m) be the codes over Z p e and Z e p respectvely. Let G B and G D be the generator matrces of the codes B and D respectvely. Then a generator matrx for A s gven by the m n m matrx: G A = p e [pem copes of G B ] + p e [pem copes of G D ],.e., G A = p e p e m p e m copes of {T r (c l )} copes of {T r (ζ c l )}. p e m copes of {T r (ζ m c l )} + pe p e m p e m copes of {T r (c t)} copes of {T r (ζ c t)}. p e m copes of {T r (ζ m c t)} where l =,,..., p em and t =,,..., p em. We need to show that the rows of G A are lnearly ndependent and generate A. Ths s easy to see snce the th row of G A, 0 m can be wrtten as, (3.7) x = p e [ T r (ζ c l ) ] + p e [ T r (ζ c t ) ], where l ranges from to p em and t ranges from to p em. Clearly the x are lnearly ndependent n m -tuples over Z n, snce the ζ are lnearly ndependent n GR(p e, m), =,, and the T r are surjectve and not dentcally zero. In addton the code A can be generated by tang all the lnear combnatons of the rows of G A. If we consder the rows of A as codewords over Z n then from Theorem 3.3 the Hammng weght of each nonzero codeword s gven by (n p p j )nm, where = 0,,,..., e and j = 0,,,..., e. If p > p and e e, the mnmum Hammng weght s gven by (n p e pe )n m. Snce A s a lnear code the mnmum Hammng dstance d H = (n p e pe )n m. Thus [n,, d H ] = [ n m, m, (n p e pe )n m ]. Example 3.6. In ths example we llustrate the code constructed by usng the trace-le map over R(6, ) = GF (, ) GF (3, ). Let T be the trace-le map over R(6, ), T r be the trace map over GF (, ) and T r the trace map over GF (3, ). The code over GF (, ) obtaned va the trace map T r s: B = [T r (a b )] a,b GF (,) = ; and G B = [ ]

10 0 N. PINNAWALA AND A. RAO s the generator matrx. Whereas the code over GF (3, ) obtaned va the trace map T r s: D = [T r (a b )] a,b GF (3,) = whch has generator matrx: [ G D = G A = 3 [ 9 copes of G B ] + [4 copes ] of[ G D ] ] = 3 + [ ] = s a generator matrx for the code A = [T (ab)] a,b R(6,) wth parameters [36,, 8] gven n Fgure below. It s relatvely straght forward to extend these results to the case n = = pe. Theorem 3.7. Let T r be the trace map over GR(p e, m), =,..., as defned n secton. The mappng defned over R(n, m) by = T : R(n, m) Z n n T (c) = T r (c ) = p e satsfes the followng propertes: For any c, c R(n, m) and α Z n. T (c + c ) = T (c) + T (c ). T (αc) = αt (c). T s surjectve v. For any c R(n, m), as x ranges over R(n, m), T (cx) taes each element n { } n (3.8) S l = p l t t = 0,,,..., = pl equally often,,...,. = pl Theorem 3.8. Let ω = exp nm tmes, where l = (l, l,..., l ), 0 l e for = ]. ( ) π n be the complex n th root of unty, where n = = pe, and C n be the set of all complex nth root of unty. The set mappng

11 COCYCLIC BUTSON HADAMARD MATRICES AND CODES VIA THE TRACE MAP Fgure. Code A = [T (ab)] a,b R(6,) wth parameters [36,, 8] ϕ : R(n, m) R(n, m) C n ϕ(a, b) = ω T (ab) s a cocycle. The matrx M ϕ = [ϕ(a, b)] a,b R(n,m) s a Butson Hadamard matrx of order n m. The rows of the exponent matrx assocated wth M ϕ (.e., A = [T (ab)] for a, b R(n, m)) form a lnear code over Z n wth parameters [n, ] = [n m, m]. In the case p < p <... < p and e e... e, the mnmum Hammng weght s gven by d H (n p e pe... pe )n m. The generator matrx G A of the code A s gven by

12 N. PINNAWALA AND A. RAO G A = ( ) [( n n = p e p e ) m copes of G ], [ ] where G s the generator matrx of the code A = T r (cc ). c,c GR(p e,m) Note that each row of G A contans the elements of Z n equally often n m tmes. In the case n = 6, the code obtaned by the constructon above can be shown to be the senary smplex code [8]. Let G α m be a m m 3 m matrx over Z 6 consstng of all possble dstnct columns. Inductvely, G α m s wrtten as [ ] G α m= G m G m G m G m G m G m wth G α = [0345]. The code s α m generated by G α m, s called a senary smplex code, because ts codewords are equdstant wth respect to the Chnese Eucldean dstance. Thus we have shown the followng: Corollary 3.9. In the case of p =, p = 3, e = e =, the generator matrx G A s permutaton equvalent to G α m. Hence the code generated by G A s a senary smplex code of type α and n partcular ths s a cocyclc senary smplex code of type α. 4. Lee, Eucldean and Chnese Eucldean Weghts of the codewords of A Let n = = pe and A = [T (ab)] a,b R(n,m) the code defned n Theorem 3.8,(). For =,,...,, let l = (l, l,..., l ), 0 l e, n l = = pl and n l = n/n l. From Theorem 3.7(v), f x s a codeword n A, then the coordnates of x tae values n S l = {n l t t = 0,,,..., n l } equally often n l n m tmes. Then dependng upon the range of the l, the Lee (W L (x)), Eucldean (W E (x)) and the Chnese Eucldean (W CE (x)) weghts of x are as per the table below: Case I: p =, p >, Range of l Range of l n l W L(x) W E(x) W CE(x) l 0 l e 0 l e l = pl 4 nm+ n m (n +n l ) n m l = e 0 l e e = pl n m (n n l ) 4 n m (n n l ) Case II: p > 0 l e 0 l e =0 pl n m (n n l ) 4 n m (n n l ) n m

13 COCYCLIC BUTSON HADAMARD MATRICES AND CODES VIA THE TRACE MAP 3 5. The Weghted-Trace map So far we have studed the trace-le map and ts fundamental propertes parallel to the trace maps over Galos rngs and Galos felds. The rng R(n, m) was the drect product of Galos rngs and Galos felds of the same degree (say m). It s farly straght forward to extend ths noton to the rng R(d, n) constructed by tang the drect product of Galos rngs and Galos felds of dfferent degrees (say m, m,..., m ). Here d = p em p em... p e m and n = p e pe... pe. Let GR(p e, m ) be the Galos rng of characterstc p e and degree m, where =,,...,. Let R(d, n) be the drect product of these rngs..e., R(d, n) = GR(p e, m ) GR(p e, m )... GR(p e, m ), where d = p em p em... p e m and n = p e pe... pe. Any element c R(d, n) can be wrtten as c = (c, c,..., c ), we can wrte where c GR(p e, m ), for =,,...,. Snce GR(p e, m ) = Z m p e c as an m - tuple over Z e p..e., c = (c, c,..., cm j =,,..., m. Let M = M-tuples c = ((c, c,..., c m ), (c, c,..., c m..., c m )), where cj Z p e ), where c j Z e p, for = m. We can now wrte the elements of R(d, n) as ),..., (c, c,, for j {,,..., m }. Let c, c R(d, n) and defne the addton and multplcaton of c, c as follows: c + c = (c + c, c + c,..., c + c ) and cc = (c c, c c,..., c c ). It s easy to show that R(d, n) s a rng under these bnary operatons and also that the number of elements of R(d, n), denoted by d s gven by d = = pem,.e., d =, m ), where GR(p e, m ) s the number of elements of = GR(pe GR(p e, m ). Defnton 5. (Weghted-trace map). [9] Let T r be the trace map over the Galos rng GR(p e, m ), where =,,...,. The weghted-trace map over the rng R(d, n) s defned by T w : R(d, n) Z n n T w (x) = T r (x ). = As n Theorem 3. we can prove that the weghted-trace map satsfes the followng propertes: Theorem 5.. Let T w be the weghted-trace map over the rng R(d, n), where d = p em p em... p e m and n = p e pe... pe. For c, c R(d, n) and α Z n the followng propertes are satsfed by T w : () T w (c + c ) = T w (c) + T w (c ). () T w (αc) = αt w (c). () T w s surjectve. The weghted-trace map T w also satsfes the followng property whch s very smlar to that of the trace-le map n Theorem 3.3. Theorem 5.3. Let c = (c, c ) R(d, n) and T w be the weghted-trace map over R(d, n) as above. As x ranges over R(d, n), T w (cx) taes each element n S l = { = pl t t = 0,,,..., n l } equally often.e., dn l /n tmes, where for =,,...,, l = (l, l,..., l ), 0 l e, n l = = pl, and n l = n/n l. p e

14 4 N. PINNAWALA AND A. RAO We use T w to construct cocyclc Butson Hadamard matrces of order d and consequently to construct non-lnear codes over Z n as follows: Theorem 5.4. Let n = = pe and ω n = e π n be the complex n th root of unty. Let C n be the multplcatve group of all complex n th roots of unty and T w be the weghted-trace map over the rng R(d, n) as defned above. Then () The set mappng defned by ϕ : R(d, n) R(d, n) C n ϕ(a, b) = ω Tw(ab) n s a cocycle. () Matrx H w = [ϕ(a, b)] a,b R(d,n) s a Butson Hadamard matrx of order d. () The exponent matrx of H w,.e., A w = [T w (ab)] a,b R(d,n) forms a non-lnear code over Z n wth the parameters (d, N, w H ), where d = = pem s the length of the code, N = = pem s the number of codewords and w H = d(p )/p ) s the mnmum Hammng weght provded that p e < p e <... < p e and m < m <... < m. Proof: () and () are smlar to that of Theorem 3.5. () Snce the number of elements n R(d, n) s d, t s clear that the length of the code A w s d = = pem and the number of codewords n A w, N, s also = = pem = d. From Theorem 5.3 t s clear that the Hammng weght of each codeword n A w s gven by d = pe(m )+l, where 0 l e for =,,...,. When p e < pe <... < pe and m < m <... < m the mnmum Hammng weght of codewords n A w s w H = d p e m... p em p em = d(p )/p. Thus A w s a (d, d, d(p )/p ) code over Z n. The next example llustrates ths result. Example 5.5. Consder the rng R(, 6) = GF (, ) GF (3, ). The trace maps T r and T r over GF (, ) and GF (3, ) are gven by T r : GF (, ) Z and T r : GF (3, ) Z 3 T r (c ) = c + c T r (c ) = c respectvely. The followng tables llustrate the values of trace maps. c T r (c ) c T r (c ) 0 0 The weghted-trace map T w over the rng R(, 6) s where c GF (, ) and c GF (3, ). T w : R(, 6) Z 6 T w (c) = 3T r (c ) + T r (c ),

15 COCYCLIC BUTSON HADAMARD MATRICES AND CODES VIA THE TRACE MAP 5 The elements of the rng R(, 6) and ther weghted-trace values are gven n the followng table. c T w (c) c T w (c) (0, 0), 0 0 (0, ), 0 3 (0, 0), (0, ), 5 (0, 0), 4 (0, ), (, 0), 0 0 (, ), 0 3 (, 0), (, ), 5 (, 0), 4 (, ), The code A w = [T w (ax)] a,x R(,6) s A w = and ts parameters (d, N, w H ) are (,, 6) Clearly A w s a non-lnear code snce the sum of the 0 th and th rows s not a codeword n A w. 6. Concluson In ths paper we ntroduced a new map, the trace-le map and n general the weghted-trace map, to construct Butson Hadamard matrces and consequently to construct lnear and non-lnear cocyclc codes over Z n for n = p e pe and more generally for n = p e pe... pe. References. A. Balga, New self-dual codes from cocyclc Hadamard matrces, J. Combn. Maths. Combn. Comput. 8 (998), A. Balga and K. J. Horadam, Cocyclc Hadamard matrces over Z t Z, Australas. J. Combn. (995), J. T. Blacford and D. K. Ray-Chaudhur, A transform approach to permutaton group of cyclc codes over Galos rngs, IEEE Trans. Info. Theory 46 (000), no. 7, A. T. Butson, Generalsed Hadamard matrces, Proc. Amer. Math. Soc. 3 (96), A. B. Calderban and N. J. A. Sloane, Modular and p - adc cyclc codes, Desgns Codes and Crypto 6 (995), D. A. Drae, Partal geometres and generalsed Hadamard matrces, Canad. J. Math. 3 (979), A. J. Gareth and J. J. Mary, Elementary number theory, Sprnger-Verlag, M. K. Gupta, D. G. Glynn, and T. A. Gullver, On some quaternary self orthogonal codes, Appled Algebra, Algebrac Algorthms and Error-Correctng Codes; AAECC-4 (S. Boztas and I. E. Shparlns, eds.), Lecture Notes n Computer Scence LNCS 7, Sprnger, 00, pp..

16 6 N. PINNAWALA AND A. RAO 9. K. J. Horadam and A. Rao, Fourer transforms from a generalsed trace map, 006 IEEE Internatonal Symposum on Informaton Theory (Seattle, U.S.A.), July 006, pp R. Ldl and H. Nederreter, Fnte felds, Cambrdge Unversty press, B. McDonald, Fnte rngs wth dentty, Marcel Deer, New Yor, N. Pnnawala and A. Rao, Cocyclc smplex codes of type α over Z 4 and Z s, IEEE Trans. Info. Theory 50 (004), no. 9, A. Rao and N. Pnnawala, New lnear codes over Z p s va the trace map, 005 IEEE Internatonal Symposum on Informaton Theory (Adelade, Australa), 4-9 September 005, pp Z. X. Wan, Lectures on fnte felds and Galos rngs, World Scentfc, New Jersey, 003. School of Mathematcal and Geospatal Scences,, RMIT Unversty, GPO Box 476V,, Melbourne VIC - 300, Australa E-mal address: nmalsr.pnnawala@rmt.edu.au School of Mathematcal and Geospatal Scences,, RMIT Unversty, GPO Box 476V,, Melbourne VIC - 300, Australa E-mal address: asha@rmt.edu.au