MacWilliams type identities for some new m-spotty weight enumerators over finite commutative Frobenius rings

Transcription

1 MacWilliams ype ideniies for some new m-spoy weigh enumeraors over finie commuaive Frobenius rings Minjia Shi School of Mahemaical Sciences of Anhui Universiy, Hefei, Anhui, China Absrac Pas few years have seen an exensive use of RAM chips wih wide I/O daa (e.g. 16, 32, 64 bis) in compuer memory sysems. These chips are highly vulnerable o a special ype of bye error, called an m-spoy bye error, which can be effecively deeced or correced using bye errorconrol codes. The MacWilliams ideniy provides he relaionship beween he weigh disribuion of a code and ha of is dual. This paper inroduces m-spoy Hamming weigh enumeraor, join m-spoy Hamming weigh enumeraor and spli m-spoy Hamming weigh enumeraor for bye error-conrol codes over finie commuaive Frobenius rings as well as m-spoy Lee weigh enumeraor over an infinie family of rings. In addiion, MacWilliams ype ideniies are also derived for hese enumeraors. keywords: Bye error-conrol codes; m-spoy bye error; MacWilliams ideniy MSC(2000) 94B05, 94B15 1 Inroducion The error-conrol codes have a significan role in improving reliabiliy in communicaions and compuer memory sysem [4]. For he pas few years, here has been an increased usage of highdensiy RAM chips wih wide I/O daa, called a bye, in compuer memory sysems. These chips are highly vulnerable o muliple random bi errors when exposed o srong elecromagneic waves, radio-acive paricles or high-energy cosmic rays. To overcome his, a new ype of bye error known as spoy bye error has been inroduced in which he error occurs a random or fewer bis wihin a b-bi bye [16], if more han one spoy bye error occur wihin a b-bi bye, hen i is known as a muliple spoy bye error or m-spoy bye error [15]. To deermine he error-deecing and errorcorrecing capabiliies of a code, some special ypes of polynomials, called weigh enumeraors, are sudied. One of he mos celebraed resuls in he coding heory is he MacWilliams ideniy ha describes how he weigh enumeraor of a linear code and he weigh enumeraor of he dual code relae o each oher. This ideniy has found widespread applicaion in he coding heory [5], especially in he sudy of self-dual codes [2]. Recenly various weigh enumeraors wih respec o m-spoy Hamming (Lee) weigh have been inroduced and sudied. Suzuki e al.[15] defined Hamming weigh enumeraor for binary bye error-conrol codes, and proved a MacWilliams ideniy for i. Özen and Siap [6] and Siap [13] exended his resul o arbirary finie fields and o he ring F 2 +uf 2 wih u 2 = 0, respecively, he laer resuls were generalized o F 2 +uf 2 + +u m 1 F 2 wih u m = 0 by Shi [12]. Siap [14] defined m-spoy Lee weigh and m-spoy Lee weigh enumeraor of bye error-conrol codes over Z 4 and derived a MacWilliams ideniy. Sharma e al. inroduced join m-spoy weigh enumeraors of wo bye error-conrol codes over he ring of inegers modulo l and over arbirary finie fields wih respec o m-spoy Hamming weigh [8], m-spoy Lee weigh [9] and r-fold join m-spoy weigh [10]. Özen and Siap [7] proved a MacWilliams ideniy for he m-spoy RT weigh enumeraors of binary codes, which was generalized o he case of finie commuaive Frobenius rings by Shi [11]. Bu MaWilliams ype ideniies for m-spoy Hamming (Lee) weigh enumeraors over finie commuaive Frobenius ring has no been considered o he bes of our knowledge. Throughou his paper, we le ring R k be a finie commuaive Frobenius ring or an infinie family of rings (see Example 2.4). The organizaion of his paper is as follows: In Secion 2, we sae some preliminaries which we need o prove our main resuls. Secion 3 presens a MacWilliams The paper was submied for reviewing on 30h March. This research is suppored by NNSF of China ( , ), Talens youh Fund of Anhui Province Universiies (2012SQRL020ZD). The auhor is wih he School of Mahemaical Sciences, Anhui Universiy, Anhui , China, Associae Professor. ( smjwcl.good@163.com). 1

2 ideniy for m-spoy Hamming weigh enumeraor over R k. Secion 4 and Secion 5 deermine MacWilliams ideniies for join m-spoy weigh enumeraor and spli m-spoy Hamming weigh enumeraor over R k, respecively. Finally, we presen a MacWilliams ideniy for m-spoy Lee weigh enumeraor over R k in Secion 6. We also illusrae our resuls wih some examples. 2 Preliminaries In his secion, we begin by giving some basic definiions ha we need o derive our resuls. Le R k be a finie commuaive Frobenius ring wih uniy and N be a posiive ineger. Le us recall some basic knowledge abou R k as described in [3]. The finie commuaive ring R k is called a Frobenius ring if R k is self-injecive (i.e., he regular module is injecive), or equivalenly, (C ) = C for any submodule C of any free R k -module Rk n, where C denoes he orhogonal submodule of C wih respec o he usual Euclidean inner produc on Rk n. Moreover, in his case, C = R k n for any submodule C of Rk n, where denoes he cardinaliy of C. This is one of he reasons why only finie Frobenius rings are suiable for coding alphabes. The reader may refer o [17,18] for more deails on Frobenius rings. Remark 2.1. Many well-known finie rings are finie Frobenius rings. Here are a number of examples of finie commuaive Frobenius rings wih heir generaing characers. (i) Le R k = F be a finie field. A generaing characer χ on R k is given by χ(x) = ξ T r(x), where ξ = e 2πi p and Tr : F l F p is he race funcion from F o F p. (ii) Le R k = Z l. Se ξ = e 2πi l. Then χ(x) = ξ x, x Z l, is a generaing characer. (iii) The finie direc sum of Frobenius rings is Frobenius. If R 1,, R n each has generaing characers χ 1,, χ n, hen R k = R i has generaing characer χ = χ i. (iv) A Galois ring R k = GR(p n, r) = Z p n[x]/ f is a Galois exension of Z p n of degree r, where f is a monic irreducible polynomial in Z p n[x] of degree r. Any elemen a of R k is represened by a unique polynomial r = r 1 a ix i, wih a i Z p n. Se ξ = e 2πi p n. Then χ(a) = ξ ar 1. (v) Le R k be a finie chain ring wih maximal ideal u and le is residue field R k / u be F p n, i.e, R k = F q +uf q + +u k 1 F q. Any elemen r of R k is represened by a unique polynomial r = r 1 a iu i, wih r i F q. Se ξ = e 2πi q. Then χr = ξ ar 1. Le (G, +) be a finie abelian group and V be a vecor space over he complex numbers. The se Ĝ of all characers of G forms an abelian group under poinwise miliplicaion. For any funcion f : G V, define is Fourier ransform ˆf : Ĝ V by f(π) = x G π(x)f(x), π Ĝ. Given a subgroup H G, define an annihilaor (Ĝ : H) = {π Ĝ : π(h) = 1}. Moreover, we have Ĝ : H) = G / H. The Poission summaion formula relaes he sums of a funcion over a subgroup o he sum of is Fourier ransform over he annihilaor of he subgroup. The following lemma can be found in [18], which plays an imporman role in deriving he MacWilliams ideniy for m-spoy Hamming weigh enumeraor. Lemma 2.2. (Poisson Summaion Formula) Le H G be a subgroup, and le f : G V be any funcion from G o a complex vecor space V. Then 1 f(x) = (Ĝ : H) x H π (Ĝ:H) f(π). 2

3 Hereinafer, codes will be aken o be of lengh N where N is a muliple of bye lengh b, i.e. N = nb. Le c = (c 1, c 2,, c N ) and v = (v 1, v 2,, v N ) be wo elemens of Rk N. The inner produc of c and v, denoed by c, v, is defined as follows: Here, c i, v i = b j=1 c, v = n c i, v i = n ( b c i,j v i,j ). j=1 c i,j v i,j denoes he inner produc of c i and v i, respecively. Le C be a linear code over Rk N. The se C = {v Rk N u, v = 0, for all u C} is also a linear code over R k and i is called he dual code of C. Remark 2.3. The following is an example of a ring ha is a no a chain ring bu a finie communiaive Frobenius ring. We will use his ring o exhibi several of he resuls of he paper. Example 2.4. Le R k = F 2 [u 1, u 2,, u k ]/ u 2 i = 0, u iu j = u j u i. When k = 1, R1 = F 2 + uf 2 is a principal ideal ring. When k = 2, he ring is R 2 = F 2 +uf 2 +vf 2 +uvf 2 is a nonprincipal ideal ring and we will ofen use R 2 in he following examples. More deails abou R k can be found in [1]. The ring can also be defined recursively. Le R k = R k 1 [u k ]/ u 2 k = 0, u ku j = u j u k = Rk 1 + ur k 1. For any subse A {1, 2,, k} we will fix u A := i A u i wih he convenion ha u φ = 1. Then any elemen of R k can be represened as A {1,2,,k} c A u A, c A F 2. Since (c A ) can be hough of as a binary vecor of lengh 2 k. Le w(c A ) be he Hamming weigh of his vecor. Then χ(r k ) = ( 1) w(c A). Throughou his paper, we adop he noaions R k = l unless oherwise saed, i.e, denoe he size of finie commuaive Frobenius ring R k by l. In Example 2.4, we have l = 2 2k. 3 MacWilliams ype ideniy for m-spoy Hamming enumeraor over finie commuaive Frobenius rings In his secion, we exend he resuls in [6] and [13] o he case over finie commuaive Frobenius rings. Le us begin wih some definiions. Definiion 3.1 (see [15]). A spoy bye error is defined as or fewer bis errors wihin a b-bi bye, where 1 b. When none of he bis in a bye is in error, we say ha no spoy bye error has occurred. We can define he m-spoy weigh and he m-spoy disance over R k as follows. Definiion 3.2. Le e Rk N be an error vecor and e i Rk b be he i-h bye of e, where N = nb and 1 i n. The number of /b-errors in e, denoed by w M (e), and called m-spoy Hamming weigh is defined as w M (e) = n, where x denoes he ceiling of x for any real number i=0 wm (e i) x, i.e., he smalles ineger number greaer han or equal o x. If = 1, his weigh, defined by w M, is equal o he Hamming weigh. In a similar way, we define he m-spoy disance of wo codewords u and v as d M = n. Furher, i is also sraighforward o show ha his dh (u i,v i) 3

4 disance is a meric in R N k. Definiion 3.3. Le v = (v 1, v 2,, v b ) Rk b. Then he suppor of v is defined by supp(v)= {i v i 0} and he complemen of supp(v) is denoed supp(v). Definiion 3.4. Le c = (c 1, c 2,, c b ) Rk b and define S p (c) = {v R b k supp(v) supp(c) and p = supp(v) } and S p (c) = {v R b k supp(v) supp(c) and p = supp(v) }. The following lemmas will be needed laer when we are ready o prove our main heorem in his secion. Lemma 3.5. Le H 0 be an ideal of R k, a R k. Then we have r R k χ(ar) = { l, if a = 0, 0, if a 0. a H χ(a) = 0 and Proof. We can obain he firs asserion readily by using he definiion of characer χ. If a = 0, hen clearly χ(ar) = 1 for all r R k and hence he resul follows. Oherwise, if a is a uni, hen elemens ar, for all r R k, run over all elemens of R k, which forms a rivial ideal R k. If a is a zero divisor, hen elemens ar, for all r R k, form a proper ideal of R k. Hence, according o he firs asserion in his Lemma, if a 0, we have χ(ar) = 0. r R k Lemma 3.6. Le v = (v 1, v 2,, v b ) Rk b wih w(c) = j 0 and p {1, 2,, j}. Then χ( c, v ) = 0. 0 w(v) p supp(v) supp(c) Proof. Le {l 1, l 2,, l p } supp(c). If we define a map ϕ : R p k R k such ha ϕ(v 1, v 2,, v p ) = c l1 v c lp v p. This is a group homomorphism and he image Im(ϕ) = H is no zero since w(c) 0. Furher, H is he nonzero subgroup of R k generaed by {c l1,, c lp }. Thus, by applying he firs group isomorphism heorem, R p k / ker(ϕ) = H {0}. Le ker(ϕ) = m. ( p ) χ( c, v ) = χ c li v li = m χ(h) = 0. h H 0 w(v) p supp(v) supp(c) This proves he Lemma. (v l1,,v lp ) Similar o [12], applying he mehod of mahemaical inducion, we have he follwing lemma. Lemma 3.7. Le c = (c 1, c 2,, c b ) Rk b and w(c) 0. For all p posiive inegers, we le I p = {i 1, i 2,, i p } supp(c) and I 0 = ø. Then we have v R b k supp(v)=ip χ( c, v ) = ( 1) p. Lemma 3.8. Le c = (c 1, c 2,, c b ) Rk b and w(c) = j 0. For all 0 p j, we have ( ) j ( ) b j (i) χ( c, v ) = ( 1) p ; (ii) χ( c, v ) = (l 1) p. p p v S p(c) v S p(c) 4

5 Proof. According o Definiion 3.4 and Lemma 3.7, we ge v S p(c) χ( c, v ) = I p supp(c) supp(v)=i p χ( c, v ) = I p supp(c) ( ) j ( 1) p = ( 1) p. p Since v S p (c) wih supp(v) supp(c), we have χ( c, v ) = 1. Furher, since p = supp(v), here are ( ) b j p ways of choosing a subse of size p from he complemen of suppor of c of size p. For each subse of size p, he sum of characers equals o (l 1) p. This proves he resul. Following Lemma 3.8, we have he following corollary. Corollary 3.9. Le c = (c 1, c 2,, c b ) R b k and w(c) = j, 0 j 1 j and 0 j 2 b j. If S j1,j 2 (c) = { v R b k j 1 = supp(v) supp(c) and j 2 = supp(v) supp(c) }, hen v s j1,j 2 (c) ( )( ) j b j χ( c, v ) = ( 1) j1 (l 1) j2. j 1 j 2 Lemma Le c = (c 1, c 2,, c b ) Rk b and w(c) = j. Then v R b k Proof. Since he sum χ( c, v )z w M (v)/ = v R b k j b j ( )( ) j b j ( 1) j1 (l 1) j2 z (j1+j2)/. j 1 j 1=0 j 2=0 χ( c, v )z w M (v)/ runs over all v Rk b, we can spli he sum according o he se S j1,j 2 where j 1 and j 2 run hrough all possible cases. Hence he conclusion of his lemma follows from Corollary 3.9. Le α j = #{i : w(c i ) = j, 1 i n}. Tha is, α j is he number of byes having Hamming weigh j, 0 j b, in a codeword. The summaion of α 0, α 1,, α b is equal o he code lengh in b byes, ha is α j = n. The Hamming weigh disribuion vecor (α 0, α 1,, α b ) is deermined j=0 uniquely for he codeword c. Then, he m-spoy Hamming weigh of he codeword c is expressed as w M (c) = b j=0 j/ α j. Le A (α0,α 1,,α b ) be he number of codewords wih Hamming weigh disribuion vecor (α 0, α 1,, α b ). For example, le c = (0, u, 0, v, 0, u+v, 1, u, 1+u+uv, 0, 0, 0, 1+ u + v + uv, u, 0) be a codeword over R 2 as saed in Example 2.4 wih bye b = 3 and n = 4. Then he Hamming weigh disribuion vecor of he codeword is (α 0, α 1, α 2, α 3 ) = (1, 1, 2, 1). Therefore, A (1,1,2,1) is he number of codewords wih Hamming weigh disribuion vecor (1, 1, 2, 1). We are now ready o define he m-spoy Hamming weigh enumeraor of a bye error conrol code over R k. Definiion The Hamming weigh enumeraor for m-spoy bye error conrol code C is defined as W (z) = c C z w M (c). j 2 By using he parameer A (α0,α 1,,α b ), which denoes he number of codewords wih Hamming weigh disribuion vecor {α 0, α 1,, α b }, W (z) can be expressed as follows: W (z) = A (α0,...,α b ) (α 0,...,α b ) j=0 α 0,...,α b 0 α 0+ +α b =n b (z j/ ) αj. 5

6 Theorem Le C be a bye error conrol code over R k. The relaion beween he m-spoy /b-weigh enumeraors of C and is dual is given by W (z) = where ϑ (b,l) j (z) = j A (α 0,...,α b ) (α 0,...,α b ) j=0 α 0,...,α b 0 α 0+ +α b =n b j j 1=0 j 2=0 ( 1) j1 (l 1) j2 ( j b (z j/ ) αj = 1 j 1 )( b j j 2 ) z (j 1+j 2)/. (α 0,...,α b ) α 0,...,α b 0 α 0+ +α b =n b A (α0,...,α b ) (ϑ (b,l) j=0 j (z)) αj, Proof. Given a linear code C R n k, we apply he Poisson Summaion Formula wih G = Rn k, H = C, and V = C[z], he polynomial ring over C in one indeerminae. The firs ask is o idenify he characer-heoreic annihilaor (Ĝ : H) = ( R k n : C) wih C. Le ρ be a generaing characer of R k. We use ρ o define a homomorphism β : R k R k. For r R k, he characer β(r) R k has he form β(r)(s) = (rρ)(s) = ρ(sr) for s R k. One can verify ha β is an isomorphism of R k -modules. In paricular, w(r) =w(βr), where w(r) = 0 for r = 0, and w(r) 0 for r 0. Exend β o an isomorphism β : R n k R k n of Rk -modules, via β(x)(y) = ρ(yx), for x, y R n k. Again, w(x) = w(βx). For x R n k, β(x) ( R k : C) means β(x)(c) = β(c x) = 1. This means ha he ideal C x of R k is conained in ker(ρ). Because ρ is a generaing characer, which implies ha C x = 0. Thus x C. The converse is obvious. Thus C corresponds o ( R k : C) under he isomorphism β. Remember ha β : R k n R k n is an isomorphism of Rk -modules and (C ) = C. Thus he Poisson Summaion Formula becomes where he Fourier ransform is v C f(v) = 1 f(c) = v R N k f(c), (1) c C Le f(v) = z w M (v). Then he funcion f(c) is calculaed as follows: f(c) = χ( c, v )z w M (v) v R nb k = = v 1 Rk b v 2 Rk b v i R b k v n R b k n χ( c i, v i )z w H(v i)/. χ c (v)f(v). (2) χ( c 1, v 1 )χ( c 2, v 2 ) χ( c n, v n ) By applying Corollary 3.9, we have, n k i b k i ( )( ) f(c) = ki b ki ( 1) j1 (l 1) j2 where k i = w(c i ). Thus b f(c) = j=0 j 1=0 j 2=0 j j 1=0 j 2=0 j 1 j 2 b j ( )( ) j b j ( 1) j1 (l 1) j2 j 1 where α j (c) = {i w(c i ) = j}. n j b j ( )( ) f(c) = j b j ( 1) j1 (l 1) j2 j 1 j 1=0 j 2=0 6 j 2 j 2 n z (j1+j2)/ z (j1+j2)/ z (j1+j2)/ z w H(v i)/, α j(c) α j(c),.

7 Afer rearranging he summaions on boh sides according o he weigh disribuion vecors of codewords in C and C respecively, we have he resul W (z) = A (α 0,...,α b ) (α 0,...,α b ) j=0 α 0,...,α b 0 α 0+ +α b =n b (z j/ ) αj = 1 (α 0,...,α b ) α 0,...,α b 0 α 0+ +α b =n b A (α0,...,α b ) (ϑ (b,l) j=0 j (z)) αj. Example Le C be he bye error-conrol code over R 2 as saed in Example 2.4 generaed by (1, 0, 0, u, v, 1 + u), (0, u, 0, u + v, uv, u), (uv, 0, uv, uv, 0, uv). This code has ype (16) 1 (8) 1 (2) 1 and hence = 256. The dual code of C is a bye error-conrol code of lengh 6 over R 2 and i has 16 4 = codewords. The Hamming weigh disribuion vecors of he codewords of C, he number of codewords, and polynomials ϑ (3,16) j (z) for b = 3 and = 2 are shown in Tables I and II for he necessary compuaions o apply Theorem Table I Hamming weigh disribuion vecors of he codewords in C and he number of codewords. (α 0, α 1, α 2, α 3) number (2, 0, 0, 0) 1 (0, 2, 0, 0) 5 (0, 0, 2, 0) 26 (0, 0, 0, 2) 64 (1, 1, 0, 0) 2 (1, 0, 1, 0) 1 (1, 0, 0, 1) 1 (0, 1, 1, 0) 19 (0, 1, 0, 1) 31 (0, 0, 1, 1) 106 Table II Polynomials ϑ (3,16) j (z) for = 2, b = 3 and l = 16. ϑ (3,16) 0 (z) = z z 2 ϑ (3,16) 1 (z) = z 225z 2 ϑ (3,16) 2 (z) = 1 16z + 15z 2 ϑ (3,16) 3 (z) = 1 z 2 According o he expression of W (z) and Table I, we obain he m-spoy Hamming weigh enumeraor of C as W (z) = z + 183z z z 6. By applying Theorem 3.12 and Table II, we obain (here for convenience, we wrie ϑ (b,l) j (z) = ϑ j (z)) W (z) = 1 A (α0,α 1,α 2,α 3) α 0+α 1+α 2+α 3=2 j=0 3 (ϑ j (z)) αj = 1 [ (ϑ0 (z)) 2 + 5(ϑ 1 (z)) (ϑ 2 (z)) (ϑ 3 (z)) 2 + 2ϑ 0 (z)ϑ 1 (z) + ϑ 0 (z)ϑ 2 (z) 256 +ϑ 0 (z)ϑ 3 (z) + 19ϑ 1 (z)ϑ 2 (z) + 106ϑ 2 (z)ϑ 3 (z) + 31ϑ 1 (z)ϑ 3 (z) ] = z z z z 4. 4 MaWillians ype ideniy for join m-spoy Hamming weigh enumeraor over finie commuaive Frobenius rings 7

8 Sharma e al. defined and sudied join m-spoy Hamming weigh enumeraor for a pair of bye error-conrol codes over he ring of inegers modulo l (l is an ineger) and over arbirary finie fields in [8]. In his secion, we enend heir resuls o arbirary finie commuaive Frobenius rings. Lemma 4.1. Le be a fixed posiive ineger, and a, b be arbirary nonnegaive inegers. If ā and b, respecively, are he leas nonnegaive residues of a and b modulo, hen we have a + b = a a a + b, if ā + b = 0; + b + 1, if 0 < ā + b ; + b + 2, if < ā + b 2 2, where x denoes he floor of x for any real number x, i.e., he smalles ineger number less han or equal o x. Proof. The proof is rivial. Definiion 4.2. Define he funcions f 01, f 10 and f 11 for each pair of vecors u = (u 1, u 2,, u b ), v = (v 1, v 2,, v b ) in Rb k, as follows: f 01 is he number of j s such ha u j = 0 and v j 0; f 10 is he number of j s such ha u j 0 and v j = 0; f 11 is he number of j s such ha u j 0 and v j 0; Noe ha for each u, v R b k, hen f 10(u, v )+f 11 (u, v ) = w H (u ), f 01 (u, v )+f 11 (u, v ) = w H (v ) where w H (u ) and w H (v ) are he Hamming weighs of u and v, respecively. Definiion 4.3. We define he funcions J, K, L: Rk b Rb k Z as follows: f01(u,v ) J(u, v, if f 01 (u, v ) + f 11 (u, v ) = 0; ) = f01(u,v ) + 1, if 0 < f 01 (u, v ) + f 11 (u, v ) ; f01(u,v ) + 2, if < f 01 (u, v ) + f 11 (u, v ) 2 2, f10(u,v ) K(u, v, if f 10 (u, v ) + f 11 (u, v ) = 0; ) = f10(u,v ) + 1, if 0 < f 10 (u, v ) + f 11 (u, v ) ; f10(u,v ) + 2, if < f 10 (u, v ) + f 11 (u, v ) 2 2, L(u, v f11 (u, v ) ) = for every u, v R b k. Applying Lemma 4.1, i is easy o see ha J(u, v ) + L(u, v f01 (u, v ) + f 11 (u, v ) wh (v ) ) = = = w M (v ) and K(u, v ) + L(u, v f10 (u, v ) + f 11 (u, v ) wh (u ) ) = = = w M (u ) From he disscussion above, we can easily ge he following proposiion. Proposiion 4.4. There exis funcions: J, K, L : R bn k Rbn k J(u, v) + L(u, v) = w M (v), K(u, v) + L(u, v) = w M (u) Z, saisfying he following: for all u = (u 1, u 2,, u n ), v = (v 1, v 2,, v n ) R bn k wih u i s and v i s in R b k. Now we are ready o define join m-spoy Hamming weigh enumeraor for a pair of bye error-conrol codes over R k. 8

9 Definiion 4.5. Le C and D be bye error-conrol codes of lengh bn and bye lengh b over R k. Then he join m-spoy Hamming weigh enumeraor of he codes C and D is defined as J (C,D) (x, y, z) = x J(u,v) y K(u,v) z L(u,v). u C v D The following heorem shows ha he join m-spoy weigh enumeraor generalizes m-spoy weigh neumeraor jus like he join probabiliy densiy funcion generalizes single probabiliy densiy funcion. Theorem 4.6. The join m-spoy Hamming weigh enumeraor J C,D (x, y, z) of bye error-conrol codes C and D over R k saisfies he following properies: (i) J (C,D) (1, 1, 1) = D, (ii) J (D,C) (x, y, z) = J (C,D) (y, x, z), (iii) W C (z) = 1 D J (C,D)(1, z, z), where W C (z) is he m-spoy Hamming weigh enumeraor of C, (iv) W D (z) = 1 J (C,D)(z, 1, z), where W D (z) is he m-spoy Hamming weigh enumeraor of D. Proof. The proof is similar o ha of Theorem 12 in [8]. For our purpose, we need he following definiions. Definiion 4.7. Le, ν, µ be he inegers saisfying 1 b and 0 ν, µ b, and le δ be an ineger saisfying ν + µ b δ min{ν, µ}. For an ineger p (0 p b), le A p be he se of all 4-uple α = (α 1, α 2, α 3, α 4 ) of nonnegaive inegers α i s saisfying α 1 + α 2 + α 3 + α 4 = p wih 0 α 1 δ, 0 α 2 µ δ, 0 α 3 ν δ, 0 α 4 b + δ ν δ. Then we define he polynomial G δ ν,µ(x, y, z) as b p=0 p g p (α)x µ α 1 α 2 +θ (α) p y p α 1 α 2 +η (α) p z α 1 +α 2 where for each p (0 p b), he summaion runs over he se A p ; and furher for each α A p, p he coefficien g p (α) is given by ( )( )( )( ) δ µ δ ν δ b + δ µ ν ( 1) α1+α3 (l 1) p α1 α2. α 1 and he number θ p (α), η p (α) α 2 α 3 α 4 are given by 0, if µ α 1 α 2 + α 1 + α 2 ; 1, if 0 < µ α 1 α 2 + α 1 + α 2 ; 2, if < µ α 1 α 2 + α 1 + α 2 2 2, 0, if p α 1 α 2 + α 1 + α 2 ; 1, if 0 < p α 1 α 2 + α 1 + α 2 ; 2, if < p α 1 α 2 + α 1 + α 2 2 2, θ (α) p = η (α) p = Definiion 4.8. For each a Z 4 2, le [a] i (1 i b) denoe he i-h componen of a. Then for any 4-uple (l 1, l 2, l 3, l 4 ) over he se {0, 1, }, we define T l1l 2l 3l 4 as he se of all 4-uple a Z 4 2 saisfing [a] i = l i if l i is eiher 0 or 1; and [a] i runs over Z 2 if l i =. Definiion 4.9. Le (1 b) be an ineger and le µ, ν, δ be he inegers saisfying 0 µ, ν, δ b. For inegers p, q(0 p, q b), le B pq be he se of all uples α = (α a : a Z 4 2) of nonnegaive inegers α a s saisfying he following: α a = δ, α a = ν, α a = µ, α a = b, α a = p, α a = q. a T 1 1 a T 1 a T 1 a T 1 a T 1 9 a Z 4 2

10 Then we define he polynomial H (δ) µ,ν(x, y, z) as b p,q=0 p,q h pq (α)x q ψ pq (α) +ζ (α) pq y p ψ pq (α) +ω (α) pq z ψ pq (α), where for each p, q(0 p, q b), he summaion p,q runs over he se B pq ; and furher for each uple α B pq, he coefficien h pq (α) is given by wih φ pq (α) = α a + a T 11 δ!(ν δ)!(µ δ)!(b + δ ν µ)! ( 1) φ(α) pq (l 1) p+q φ pq(α) α a! a Z 4 2 α a, ψ pq (α) = a T 11 ζ (α) pq = ω (α) pq = a T 11 α a and he numbers 0, if q ψ pq (α) 1, if 0 < q ψ pq (α) 2, if < q ψ (α) + ψ (α) pq ; pq 0, if p ψ pq (α) 1, if 0 < p ψ pq (α) 2, if < p ψ (α) + ψ (α) pq ; + ψ (α) pq 2 2, + ψ (α) pq ; pq + ψ (α) pq ; + ψ (α) pq 2 2, If, for 1 i n, j i is he Hamming weigh of i-h bye u i = (u i1, u i2,, u ib ) of he vecor u = (u 1, u 2,, u n ) Rk bn, hen he vecor (j 1, j 2,, j n ) is called he Hamming weigh disribuion vecor of u and is denoed by w D (u). Le j = (j 1, j 2,, j n ), k = (k 1, k 2,, k n ) and δ = (δ 1, δ 2,, δ n ) be he Hamming weigh disribuion vecors of u, v and w, where u, v, w Rk bn and 0 j i, k i, δ i b for each i, we define he polynomials G (δ) (x, y, z) = n G (δi) j i,k i (x, y, z), H (δ) (x, y, z) = n H (δi) j i,k i (x, y, z). We also define j k = (j 1 k 1, j 2 k 2,, j n k n ), where j i k i = min{j i, k i } for each i. Furher, le v = (v 1, v 2,, v n ) where v i = (v i1, v i2,, v ib ) Rk b. Then we define a vecor u v Zbn 2 as u v = (u 1 v 1, u 2 v 2,, u n v n ), where for each i, u i v i = (u i1 v i1, u i2 v i2,, u ib v ib ) wih each u ij v ij given by { 1 if u ij 0 and v ij 0; u ij v ij = 0 ohwise i is easy o see ha 0 w H (u i v i ) j i k i for each i, so ha w D (u v) varies from 0 o j k. Theorem Le C and D be bye error-conrol codes of lengh bn and bye lengh b over R k. If A δ (j; k) is he number of pairs (u, v) of codewords u C and v D having w D (u) = j, w D (v) = k and w D (u v) = δ, hen we have (i) J C,D(x, y, z) = 1 (ii) J C,D (x, y, z) = 1 D j k δ=0 δ=0 1 (iii) J C,D (x, y, z) = D j k A δ (j; k)g (δ) (x, y, z), A δ (j; k)g (δ) k,j (y, x, z), j k δ=0 A δ (j; k)h (δ) (y, x, z), 10 (3) (4)

11 where he summaion runs over all n-uples j = (j 1, j 2,, j n ) and k = (k 1, k 2,, k n ) saisfying 0 j i, k i b, and he polynomials G (δ) (x, y, z) s and H(δ) (x, y, z) s are defined in Definiion 4.9. Proof. The proof is similar o hose of Theorem 3.12 and Theorem 20 in [8]. Example Le C be he bye error-conrol code over R 2 generaed by he se {(1, 0, 0, u, v, 1, 0, 0, u), (0, 0, uv, uv, 0, 0, 0, uv, uv)}. Is lengh is 9 and bye lengh is 3. I is easy o check ha he generaors are independen, hence he code has ype (16) 1 (2) 1 and = 32. Is dual code C, which is also a bye error-conrol code over R 2 of lengh 9, conains = codewords. Le D be a linear code of lengh 9 and bye lengh 3, generaed by (0, 0, uv, uv, 0, 0, 0, uv, uv). Since C = is large and D = 2 is small, we apply Theorem 4.10 (i) o obain join m-spoy weigh enumeraor of he codes C and D. For his, we need o compue Hamming weigh disribuion vecors of he codewords in C, which are given in Table IV. I is easy o see ha he codewords u = (1, 0, 0, u, v, 1, 0, 0, u) C and he unique nonzero v = (uv, 0, uv, 0, uv, 0, 0, 0, uv) D having Hamming weigh disribuion vecors as j = (1, 3, 1) and k = (2, 1, 1), respecively, have δ = (1, 1, 1) and hey conribue G (δ) (x, y, z) = G(1) 1,2 (x, y, z)g(1) (x, y, z)g(1) (x, y, z) o he join m-spoy Hamming weigh enumeraor, where by Definiion 4.7, G (1) 1,2 (x, y, z) = x+239xy 225yz 15z, G(1) 3,1 (x, y, z) = x xy2 and G (1) 1,1 (x, y, z) = x+224xy 225xy2. Working in a similar way, we obain he conribuing polynomials for each riple of j, k and δ, which are given in Table III (here for convenience, we wrie G (δ) (x, y, z) = G(δ) ). Now by Theorem 4.10 (i), he join m-spoy Hamming weigh enumeraor of C and D is given by J C,D(x, y, z) = 1 j k δ=0 A δ (j; k)g (δ) (x, y, z) 3,1 = x 3 y x 3 y x 3 y x 3 y x 3 y + x x 2 y 5 z x 2 y 4 z x 2 y 3 z x 2 y 2 z +7715x 2 yz y y y y y y ,1 5 MaWillians ype ideniy for spli m-spoy Hamming weigh enumeraor over Finie commuaive Frobenius rings In his secion, we prove ha he resuls in [8] sill hold over arbirary finie commuaive Frobenius rings. We firs recall spli m-spoy Hamming weigh enumeraor for a bye error-conrol code over R k as follows: Definiion 5.1. Le C be a bye error-conrol code of lengh bn over R k wih bye lengh b. Then he spli m-spoy Hamming weigh enumeraor of he code C, denoed by S C (x i, y i : i = 1, 2,, n), is defined as ( n ) x b/ w M (u i) i y w M (u i) i. (u 1,u 2,,u n) C We recall ha if j i (1 i n) is he Hamming weigh of he i-h bye u i of u = (u 1, u 2,, u n ), hen he vecor (j 1, j 2,, j n ) is called he Hamming weigh disribuion vecor of u. If A(j 1, j 2,, j n ) denoes he number of codewords in C having Hamming weigh disribuion vecor as 11

12 j k δ A δ (j, k) G (δ) (x, y, z) (0, 0, 0) (0, 0, 0) (0, 0, 0) 1 G (0) 0,0 G(0) 0,0 G(0) 0,0 (0, 0, 0) (2, 1, 1) (0, 0, 0) 1 G (0) 0,2 G(0) 0,1 G(0) 0,1 (1, 1, 0) (0, 0, 0) (0, 0, 0) 1 G (0) 1,0 G(0) 1,0 G(0) 0,0 (1, 1, 0) (2, 1, 1) (1, 0, 0) 1 G (1) 1,2 G(0) 1,1 G(0) 0,1 (1, 1, 2) (0, 0, 0) (0, 0, 0) 1 G (0) 1,0 G(0) 1,0 G(0) 2,0 (1, 1, 2) (2, 1, 1) (1, 0, 1) 1 G (1) 1,2 G(0) 1,1 G(1) 2,1 (1, 2, 0) (0, 0, 0) (0, 0, 0) 2 G (0) 1,0 G(0) 2,0 G(0) 0,0 (1, 2, 0) (2, 1, 1) (1, 1, 0) 2 G (1) 1,2 G(1) 2,1 G(0) 0,1 (1, 2, 1) (0, 0, 0) (0, 0, 0) 2 G (0) 1,0 G(0) 2,0 G(0) 1,0 (1, 2, 1) (2, 1, 1) (1, 0, 1) 2 G (1) 1,2 G(0) 2,1 G(1) 1,1 (1, 3, 1) (0, 0, 0) (0, 0, 0) 10 G (0) 1,0 G(0) 3,0 G(0) 1,0 (1, 3, 1) (2, 1, 1) (1, 1, 1) 10 G (1) 1,2 G(1) 3,1 G(1) 1,1 (2, 1, 1) (0, 0, 0) (0, 0, 0) 2 G (0) 2,0 G(0) 1,0 G(0) 1,0 (2, 1, 1) (2, 1, 1) (2, 0, 0) 2 G (2) 2,2 G(0) 1,1 G(0) 1,1 (2, 2, 1) (0, 0, 0) (0, 0, 0) 2 G (0) 2,0 G(0) 2,0 G(0) 1,0 (2, 2, 1) (2, 1, 1) (2, 1, 0) 2 G (2) 2,2 G(1) 2,1 G(0) 1,1 (2, 2, 2) (0, 0, 0) (0, 0, 0) 1 G (0) 2,0 G(0) 2,0 G(0) 2,0 (2, 2, 2) (2, 1, 1) (2, 0, 1) 1 G (2) 2,2 G(0) 2,1 G(1) 2,1 (2, 3, 2) (0, 0, 0) (0, 0, 0) 10 G (0) 2,0 G(0) 3,0 G(0) 2,0 (2, 3, 2) (2, 1, 1) (2, 1, 1) 10 G (2) 2,2 G(1) 3,1 G(1) 2,1 Table III Conribuing polynomials o he m-spoy Hamming weigh enumeraor G (0) 0,0 = y y2, G (0) 0,1 = x + 720xy xy2, = x + 495xy + 225z yz, G (0) 1,0 = y 225y2, G (0) 1,1 = G(1) 1,1 = x + 224xy 225xy2, G (0) 0,2 G (1) 1,2 = x + 239xy 225yz 15z, G (0) 2,0 = 1 16y + 15y2, G (0) 2,1 = G(1) G (2) 2,2 2,1 = x 16xy + 15xy2, = x + z + 15yz 17xy, G (0) 3,0 = 1 y2, G (1) 3,1 = x xy2. (j 1, j 2,, j n ), hen he spli m-spoy Hamming weigh enumeraor S C (x i, y i : i = 1, 2,, n) of he code C can be rewrien as (u 1,u 2,,u n) C A(j 1, j 2,, j n ) n x b/ ji/ i y ji/ i, where he summaion runs over all n-uples (j 1, j 2,, j n ) saisfying 0 j i b for each i, 1 i n. Theorem 5.2. Le C be a bye error-conrol code of lengh bn over R k wih bye lengh b and spli m-spoy Hamming weigh enumeraor S C (x i, y i : i = 1, 2,, n). Then he spli m-spoy Hamming weigh enumeraor of he dual code C over R k is given by S C (x i, y i : i = 1, 2,, n) = 1 (j 1,j 2,,j n) A(j 1, j 2,, j n ) n g () j i (x i, y i ), where he summaion runs over all n-uples (j 1, j 2,, j n ) saisfying 0 j i b for 1 i n, and he polynomials g () j i (x i, y i ) are definied by g () j i (x i, y i ) = where for each p, he polynomial K p (j i ) = b K p (j i )x b/ p/ y p/, (5) p=0 p a=0 ( 1) a (l 1) p a( j i a )( b ji polynomial. (Here, we assume ha ( e f) = 0 when f < 0 or f > e.) Proof. The proof is similar o hose of Theorem 3.12 and Theorem 25 in [8]. p a) is he well-known Krawchouk 12

13 Theorem 5.3. Le C, D be bye error-conrol codes of lengh bn over R k wih bye lengh b and m-spoy Hamming weigh enumeraor W C (z), W D (z) and spli m-spoy Hamming weigh enumeraors S C (x i, y i : i = 1, 2,, n), S D (x i, y i : i = 1, 2,, n), respecively. Then (i) he direc sum C D = (u v : u C, v D) has m-spoy Hamming weigh enumeraor W C (z)w D (z) and spli m-spoy Hamming weigh enumeraor S C (x i, y i : i = 1, 2,, n)s D (X i, Y i : i = 1, 2,, n). (ii) assuming n even, he code C D = {(u v u v ) : u = (u u ) C, v = (v v ) D} (where u and v have each been broken ino wo equal halves) has m-spoy Hamming weigh enumeraor W C (z)w D (z) and spli m-spoy Hamming weigh enumeraor S C (x i, y i ; X i, Y i : i = 1, 2,, n/2)s D (x i, y i ; X i, Y i : i = (n/2) + 1,, n). Proof. The proof is similar o ha of Theorem 28 in [8]. Example 5.4. Le C be he bye error-conrol code as defined in Example Here we apply Theorem 5.2 o compue spli m-spoy Hamming weigh enumeraor of he code C. For his, firs we need o compue he Hamming weigh disribuion vecors for he codewords of C. I is easy o see ha he codeword (1, 0, 0, u, v, 1, 0, 0, u) C has Hamming weigh disribuion vecor as (1, 3, 1) and i conribues he polynomial g (2) o he spli m-spoy Hamming weigh enumeraor, where by (5), we have g (2) 1 (x i, y i ) = x 2 i + 224x iy i 225yi 2, g(2) 3 (x i, y i ) = x 2 i y2 i for each i. Working similarly, we obain he Hamming weigh disribuion vecor and he conribuing polynomials for oher codewords in C, which are given in Table IV. Therefore by Theorem 5.2, he spli m-spoy Hamming weigh enumeraor of he code C is given by S C (x i, y i : i = 1, 2, 3) = 1 3 (j 1,j 2,j 3) g (2) j i (x i, y i ) = 2x 2 1x 2 2x x 2 1x 2 2x 3 y x 2 1x 2 2y x 2 1x 2 y 2 x x 2 1x 2 y 2 x 3 y x 2 1x 2 y 2 y x 2 1y 2 2x x 2 1y 2 2x 3 y x 2 1y 2 2y x 1 y 1 x 2 2x x 1 y 1 x 2 2x 3 y x 1 y 1 x 2 2y x 1 y 1 x 2 y 2 x x 1 y 1 x 2 y 2 x 3 y x 1 y 1 x 2 y 2 y x 1 y 1 y 2 2x x 1 y 1 y 2 2x 3 y x 1 y 1 y 2 2y y 2 1x 2 2x 3 y y 2 1x 2 2y y 2 1x 2 y 2 x y 2 1x 2 y 2 x 3 y y 2 1x 2 y 2 y y 2 1y 2 2x y 2 1y 2 2x 3 y y 2 1y 2 2y MacWilliams ype ideniy for m-spoy Lee weigh enumeraor over an infinie family of rings In his secion, we concenrae our sudy on m-spoy Lee weigh enumeraor for bye errorconrol codes over R k = R k 1 [u k ]/ u 2 k = 0, u ku j = u j u k. We prove ha he resuls in [9] are sill valid over R k. For his, we firs recall he definiion of Lee weighs in R k as follows: Definiion 6.1. For any subse A {1, 2,, k}, he Lee weigh of u A is defined by w L (u A ) = 2 A. In R k, here are precisely ( ) 2 k i elemens of Lee weigh i, for i = 0, 1,, 2 k. For example, consider R 2 = F 2 + uf 2 + vf 2 + uvf 2, he Lee weigh of elemens 1, 1 + u, 1 + v and 1 + u + v + uv is 1; of elemens u, v, u + v, u + uv, v + uv and u + v + uv is 2; of elemens 1 + uv, 1 + u + uv, 1 + v + uv and 1 + u + v is 3; of elemen uv is 4; of elemen 0 is 0. 13

14 Table IV Conribuing polynomials of he codewords codewords of C (j 1, j 2, j 3 ) g (2) j 1 (x 1, y 1 )g (2) j 2 (x 2, y 2 )g (2) j 3 (x 3, y 3 ) (0, 0, 0, 0, 0, 0, 0, 0, 0) (0, 0, 0) g (2) 0 (x 1, y 1 )g (2) 0 (x 2, y 2 )g (2) 0 (x 3, y 3 ) (8, 0, 0, 0, 0, 8, 0, 0, 0) (1, 1, 0) g (2) 1 (x 2, y 2 )g (2) 0 (x 3, y 3 ) (0, 0, 8, 8, 0, 0, 0, 8, 8) (1, 1, 2) g (2) 1 (x 2, y 2 )g (2) (2, 0, 0, 0, 8, 2, 0, 0, 0) (1, 2, 0) g (2) 2 (x 2, y 2 )g (2) 0 (x 3, y 3 ) (10, 0, 0, 0, 8, 10, 0, 0, 0) (1, 2, 0) g (2) 2 (x 2, y 2 )g (2) 0 (x 3, y 3 ) (4, 0, 0, 8, 0, 4, 0, 0, 8) (1, 2, 1) g (2) 2 (x 2, y 2 )g (2) (12, 0, 0, 8, 0, 12, 0, 0, 8) (1, 2, 1) g (2) 2 (x 2, y 2 )g (2) (1, 0, 0, 2, 4, 1, 0, 0, 2) (1, 3, 1) g (2) (3, 0, 0, 2, 12, 3, 0, 0, 2) (1, 3, 1) g (2) (5, 0, 0, 10, 4, 5, 0, 0, 10) (1, 3, 1) g (2) (6, 0, 0, 8, 8, 6, 0, 0, 8) (1, 3, 1) g (2) (7, 0, 0, 10, 12, 7, 0, 0, 10) (1, 3, 1) g (2) (9, 0, 0, 2, 4, 9, 0, 0, 2) (1, 3, 1) g (2) (11, 0, 0, 2, 12, 11, 0, 0, 2) (1, 3, 1) g (2) (13, 0, 0, 10, 4, 13, 0, 0, 10) (1, 3, 1) g (2) (14, 0, 0, 8, 8, 14, 0, 0, 8) (1, 3, 1) g (2) (15, 0, 0, 10, 12, 15, 0, 0, 10) (1, 3, 1) g (2) (4, 0, 8, 0, 0, 4, 0, 8, 0) (2, 1, 1) g (2) 1 (x 2, y 2 )g (2) (12, 0, 8, 0, 0, 12, 0, 8, 0) (2, 1, 1) g (2) 1 (x 2, y 2 )g (2) (6, 0, 8, 0, 8, 6, 0, 8, 0) (2, 2, 1) g (2) 2 (x 2, y 2 )g (2) (14, 0, 8, 0, 8, 14, 0, 8, 0) (2, 2, 1) g (2) 3 (x 1, y 1 )g (2) 2 (x 2, y 2 )g (2) (8, 0, 8, 8, 0, 8, 0, 8, 8) (2, 2, 2) g (2) 2 (x 2, y 2 )g (2) (1, 0, 8, 10, 4, 1, 0, 8, 10) (2, 3, 2) g (2) (2, 0, 8, 8, 8, 2, 0, 8, 8) (2, 3, 2) g (2) (3, 0, 8, 10, 12, 3, 0, 8, 10) (2, 3, 2) g (2) (5, 0, 8, 2, 4, 5, 0, 8, 2) (2, 3, 2) g (2) (7, 0, 8, 2, 12, 7, 0, 8, 2) (2, 3, 2) g (2) (9, 0, 8, 10, 4, 9, 0, 8, 10) (2, 3, 2) g (2) (10, 0, 8, 8, 8, 10, 0, 8, 8) (2, 3, 2) g (2) (11, 0, 8, 10, 12, 11, 0, 8, 10) (2, 3, 2) g (2) (13, 0, 8, 2, 4, 13, 0, 8, 2) (2, 3, 2) g (2) (15, 0, 8, 2, 12, 15, 0, 8, 2) (2, 3, 2) g (2) g (2) 0 (x, y) = x xy y 2, g (2) 1 (x, y) = x xy 225y 2, g (2) 2 (x, y) = x2 16xy + 15y 2, g (2) 3 (x, y) = x2 y 2 14

15 Now we define he m-spoy Lee weigh, he m-spoy Lee disance and he m-spoy Lee weigh enumeraor for a bye error-conrol code over R k. Definiion 6.2. For any u = (u 1, u 2,, u n ) Rk bn, he m-spoy Lee weigh of u is defined as w ML (u) = n, where u i = (u i1, u i2,, u ib ) Rk b is he i-h bye of u and w L (u i ) = b j=1 w L (u ij ). wl (u i) Definiion 6.3. Le u, v be vecors in Rk bn wih heir i-h byes as u i, v i respecively. Then he m-spoy Lee disance beween u and v, denoed by d ML, is defined as d ML = n dl (u i, v i ) wl (u i v i ) = = w ML (u v). Noe ha d ML is a meric on Rk bn. Furher, if C is a bye error-conrol code of lengh bn and bye lengh b over R k, hen he number d ML (C) = min{d ML (u, v) : u, v C, u v} is called he m-spoy Lee disance of he code C. Moreover, d ML (C) = min{w ML (u) : u C, u 0}. Definiion 6.4. Le C be a bye error-conrol code of lengh bn and bye lengh b over R k. Then he m-spoy Lee weigh enumeraor of C is defined as L C (z) = u C z w ML(u) = u=(u 1,u 2,,u n) C j=1 n z w L(u i)/. Le u be any vecor in Rk bn wih u i Rk b as is i-h bye for 1 i n. For each i, if j ip(0 p l 1) is he number of bis in u i which are equal o r p, hen he l-uple J i = (j i0, j i1,, j i,l 1 ) is called composiion of he i-h bye u i of u, and he vecor J = (J 1, J 2,, J n ) is called he composiion vecor of u = (u 1, u 2,, u n ). Now le A(J ) be he number of codewords in C having he composiion vecor as J. Then he m-spoy Lee weigh enumeraor of C can be rewrien as where for each i, J i L C (z) = J A(J ) n z ρ(j i)/, j=1 = (j i0, j i1,, j i,l 1 ) and ρ(j i ) = l 1 p=0 w L (r p )j ip wih 0 j ip b for 0 p l 1. Le u be he fixed vecor. Define he numbers s pq, 0 p, q l 1, as he number of componens of he vecors u = (u 1, u 2,, u b ) and v = (v 1, v 2,, v b ) saisfying u i = r p and v i = r q. Definiion 6.5. For a fixed posiive ineger and J = (j 0, j 1,, j l 1 ), we define he polynomial g () J (z) = ( l 1 l 1 ) ( l 1 j p! l 1 χ r l 1 p r q s pq z w L (r q)s pq )/ p=0 q=1, s pq p=0 s pq! q=0 q=0 where he summaion s pq runs over all non-negaive inegers s pq (0 p, q l 1) saisfying l 1 q=0 s pq = j p for every p. Lemma 6.6. For a fixed vecor u Rk b wih composiion J = (j 0, j 1,, j l 1 ), we have χ( u, v )z wl(v) = g () J (z). v R b k 15

16 Proof. The proof is similar o ha of Lemma 2 in [9]. In he following heorem, we derive a MacWilliams ype ideniy for he m-spoy Lee weigh enumeraor of a bye error-conrol code over an infinie family of rings. Theorem 6.7. Le C be a bye error-conrol code of lengh bn over R k wih bye lengh b and m-spoy Lee weigh enumeraor L C (z) as defined above. If A(J 1, J 2,, J n ) denoes he number of codewords in C having he composiion vecor as J = (J 1, J 2,, J n ), hen he m-spoy Lee weigh enumeraor L C of he dual code C is given by L C (z) = 1 n A(J ) J g () J i (z), where he summaion runs over all n-uples J wih each J i, an l-uple over {0, 1, 2,, b}. Proof. The proof is similar o hose of Theorem 3.12 and Theorem 28 in [9]. Example 6.8 Le C be he bye error-conrol code as defined in Example Since C = is very large, we will apply Theorem 6.7 o obain he m-spoy Lee weigh enumeraor of C. For his, we compue he composiion vecors for he codewords of C. I is easy o check ha (0, 0, uv, uv, 0, 0, 0, uv, uv) C and is composiion vecor is J 1, J 2 and J 3 where (J 1, J 2, J 3 ) = (( [2] 0 [1] 8 ), ( [2] 0 [1] 8 ), ( [1] 0 [2] 8 )). Here, [s] p represens here are s elemens equal o r p, where 0 p 15, 0 s 3 and r p R 2 is defined in Table V, which gives an alernaive expression form for every elemen r p = auv + bv + cu + d R 2, where (a, b, c, d) Z 4 2. For = 2, by Definiion 6.5, we ge g (2) J 1 (z) = g (2) J 2 (z) = 1 + 6z 29z z 3 9z 4 10z 5 + 5z 6 and g (2) J 3 (z) = 1 2z 5z z 3 25z z 5 3z 6. We noe ha he codeword (0, 0, uv, uv, 0, 0, 0, uv, uv) C conribues he erm g (2) J 1 (z)g (2) J 2 (z)g (2) J 3 (z) = z 51z 2 272z z z z z z z z z z z z z z z 17 75z 18 o he m-spoy Lee weigh enumeraor of C. Working similarly, we obain he composiion vecors and he conribuing polynomials for oher codewords in C, which are given in Table VI. Therefore by Theorem 6.7, he m-spoy Lee weigh enumeraor of he code C is given by L C = 1 n A(J ) J g () J i (z) = 101z z z z z z z z z z z z z z z z z z Conclusion This paper mainly presens he MacWilliams ype ideniies for he m-spoy Hamming weigh enumeraor, join m-spoy Hamming weigh enumeraor and spli m-spoy Hamming weigh enumeraor for bye error-conrol codes over finie commuaive Frobenius rings. Finally, he m-spoy Lee weigh enumeraor over an infinie family of rings R k is also obained. In fac, Join m-spoy Lee weigh enumeraor (Theorem 8 and Theorem 9 in [9]) and Spli m-spoy Lee weigh enumeraor (Theorem 5 in [9]) are also valid over he infinie family of rings R k. Moreover, he resuls in [10] for r-fold join m-spoy Hamming weigh enumeraors sill hold over finie commuaive Frobenius rings by applying he similar mehod in Theorem

17 Table V Alernaive expression of elemens of R k (a, b, c, d) r p (0, 0, 0, 0) r 0 = 0 (0, 0, 0, 1) r 1 = 1 (0, 0, 1, 0) r 2 = u (0, 0, 1, 1) r 3 = 1 + u (0, 1, 0, 0) r 4 = v (0, 1, 0, 1) r 5 = 1 + v (0, 1, 1, 0) r 6 = u + v (0, 1, 1, 1) r 7 = 1 + u + v (1, 0, 0, 0) r 8 = uv (1, 0, 0, 1) r 9 = 1 + uv (1, 0, 1, 0) r 10 = u + uv (1, 0, 1, 1) r 11 = 1 + u + uv (1, 1, 0, 0) r 12 = v + uv (1, 1, 0, 1) r 13 = 1 + v + uv (1, 1, 1, 0) r 14 = u + v + uv (1, 1, 1, 1) r 15 = 1 + u + v + uv Table VI Codewords, composiion vecors and conribuing polynomials codewords (J 1, J 2, J 3) g (2) J 1 g (2) J 2 g (2) J 3 (0, 0, 0, 0, 0, 0, 0, 0, 0) (([3] 0 ), ([3] 0 ), ([3] 0 )) a 3 (8, 0, 0, 0, 0, 8, 0, 0, 0) (([2] 0 [1] 8 ), ([2] 0 [1] 8 ), ([3] 0 )) ab 2 (0, 0, 8, 8, 0, 0, 0, 8, 8) (([2] 0 [1] 8 ), ([2] 0 [1] 8 ), ([1] 0 [2] 8 )) b 2 c (2, 0, 0, 0, 8, 2, 0, 0, 0) (([2] 0 [1] 2 ), ( [3] 5 ), ([1] 0[2] 5 )) ade (10, 0, 0, 0, 8, 10, 0, 0, 0) (([2] 0 [1] 10 ), ([1] 0 [1] 8 [1] 10 ), ( [3] 2 ), ([3] 0 )) ade (4, 0, 0, 8, 0, 4, 0, 0, 8) (([2] 0 [1] 4 ), ([1] 0 [1] 4 [1] 8 ), ([2] 0 [1] 8 )) bde (12, 0, 0, 8, 0, 12, 0, 0, 8) (([2] 0 [1] 12 ), ([1] 0 [1] 8 [1] 10 ), ([2] 0 [1] 8 )) ade (1, 0, 0, 2, 4, 1, 0, 0, 2) (([2] 0[1] 1 ), ( [1] 1[1] 2 [1] 4 ), ([2] 0 [1] 2 )) cdf (3, 0, 0, 2, 12, 3, 0, 0, 2) (([2] 0 [1] 3 ), ( [1] 2[1] 3 [1] 12 ), ([2] 0 [1] 2 )) cdf (5, 0, 0, 10, 4, 5, 0, 0, 10) (([2] 0 [1] 5 ), ( [1] 4[1] 5 [1] 10 ), ([2] 0 [1] 10 )) cdf (6, 0, 0, 8, 8, 6, 0, 0, 8) (([2] 0 [1] 8 ), ([1] 6 [2] 8 ), ([2] 0 [1] 8 )) bdg (7, 0, 0, 10, 12, 7, 0, 0, 10) (([2] 0 [1] 7 ), ( [1] 7 [1] 10 [1] 12 ), ([2] 0 [1] 10 )) deg (9, 0, 0, 2, 4, 9, 0, 0, 2) (([2] 0 [1] 9 ), ( [1] 2 [1] 4 [1] 9 ), ([2] 0 [1] 2 )) deg (11, 0, 0, 2, 12, 11, 0, 0, 2) (([2] 0 [1] 11 ), ( [1] 2 [1] 11 [1] 12 ), ([2] 0 [1] 2 )) deg (13, 0, 0, 10, 4, 13, 0, 0, 10) (([2] 0 [1] 13 ), ( [1] 4 [1] 10 [1] 13 ), ([2] 0 [1] 10 )) deg (14, 0, 0, 8, 8, 14, 0, 0, 8) (([2] 0 [1] 14 ), ( [2] 8 [1] 14 ), ([2] 0 [1] 8 )) bdg (15, 0, 0, 10, 12, 15, 0, 0, 10) (([2] 0 [1] 15 ), ( [1] 10 [1] 12 [1] 15 ), ([2] 0 [1] 10 )) dcf (4, 0, 8, 0, 0, 4, 0, 8, 0) (([1] 0 [1] 4 [1] 8 ), ([2] 0 [1] 4 ), ([2] 0 [1] 8)) bde (12, 0, 8, 0, 0, 12, 0, 8, 0) (([1] 0 [1] 8 [1] 12 ), ([2] 0 [1] 12 ), ([2] 0 [1] 8)) bde (6, 0, 8, 0, 8, 6, 0, 8, 0) (([1] 0 [1] 6 [1] 8 ), ([1] 0 [1] 6 [1] 8 ), ([2] 0 [1] 8)) be 2 (14, 0, 8, 0, 8, 14, 0, 8, 0) (([1] 0 [1] 8 [1] 14 ), ([1] 0 [1] 8 [1] 14 ), ([2] 0 [1] 8)) be 2 (8, 0, 8, 8, 0, 8, 0, 8, 8) (([1] 0 [2] 8 ), ([1] 0 [2] 8 ), ([1] 0 [2] 8 )) c 3 (1, 0, 8, 10, 4, 1, 0, 8, 10) (([1] 0[1] 1 [1] 8 ), ( [1] 1 [1] 4 [1] 10 ), ([1] 0 [1] 8 [1] 10 )) c 2 e (2, 0, 8, 8, 8, 2, 0, 8, 8) (([1] 0 [1] 2 [1] 8 ), ( [1] 2 [2] 8 ), ([1] 0 [2] 8 )) ceg (3, 0, 8, 10, 12, 3, 0, 8, 10) (([1] 0 [1] 3 [1] 8 ), ( [1] 3 [1] 10 [1] 12 ), ([1] 0 [1] 8 [1] 10 )) c 2 e (5, 0, 8, 2, 4, 5, 0, 8, 2) (([1] 0 [1] 5 [1] 8 ), ( [1] 2 [1] 4 [1] 5 ), ([1] 0 [1] 2 [1] 8 )) c 2 e (7, 0, 8, 2, 12, 7, 0, 8, 2) (([1] 0 [1] 7 [1] 8 ), ( [1] 2 [1] 7 [1] 12 ), ([1] 0 [1] 2 [1] 8 )) e 3 (9, 0, 8, 10, 4, 9, 0, 8, 10) (([1] 0 [1] 8[1] 9 ), ( [1] 4 [1] 9 [1] 10 ), ([1] 0 [1] 8 [1] 10 )) e 3 (10, 0, 8, 8, 8, 10, 0, 8, 8) (([1] 0 [1] 8 [1] 10 ), ( [2] 8 [1] 10 ), ([1] 0 [2] 8 )) ceg (11, 0, 8, 10, 12, 11, 0, 8, 10) (([1] 0 [1] 8 [1] 11 ), ( [1] 10[1] 11[1] 12 )), ([1] 0 [1] 8 [1] 10 )) e 3 (13, 0, 8, 2, 4, 13, 0, 8, 2) (([1] 0 [1] 8 [1] 13 ), ( [1] 2 [1] 4 [1] 13 )), ([1] 0 [1] 2 [1] 8 )) e 3 (15, 0, 8, 2, 12, 15, 0, 8, 2) (([1] 0 [1] 8 [1] 15 ), ( [1] 2 [1] 12 [1] 15 )), ([1] 0 [1] 2 [1] 8 )) c 2 e a = z + 715z z z z z 6, b = 1 + 6z 29z z 3 9z 4 10z 5 + 5z 6, c = 1 2z 5z z 3 25z z 5 3z 6, d = z + 55z 2 132z 3 33z z 5 + 9z 6, e = 1 6z + 15z 2 20z z 4 6z 5 + z 6, f = z + 275z z 3 297z 4 154z 5 11z 6. 17

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