Generalized Reed-Muller codes over Galois rings
Harinaivo Andriatahiny, Desiré Arsène Ratahirinjatovo, Sanni José Andrianalisefa
aa r X i v : . [ c s . I T ] J a n Generalized Reed-Muller codes overGalois rings
Harinaivo ANDRIATAHINY (1) e-mail: [email protected]é Arsène RATAHIRINJATOVO (2) e-mail: [email protected] José ANDRIANALISEFA (3) e-mail: [email protected] , , Mention: Mathematics and Computer Science,Domain: Sciences and Technologies,University of Antananarivo, MadagascarOctober 12, 2018
Abstract
Recently, Bhaintwal and Wasan studied the Generalized Reed-Mullercodes over the prime power integer residue ring. In this paper, we give ageneralization of these codes to Generalized Reed-Muller codes over Galoisrings.
Keywords: Reed-Muller code, Galois ring, Code over ringMSC 2010: 94B05, 94B15, 12E05
In 1994, Hammons et al.[8] showed that some non-linear binary codes with verygood parameters are images, under the Gray map, of some linear codes over Z ,the ring of integers modulo . This led to the active study of codes over rings.Many properties of these codes are discovered. And various aspects of codingare dealt in the general setting of Galois rings instead of finite fields.Linear codes over Z p s have been studied by several authors. Borges et al.[7]defined the Reed-Muller code over Z , called the Quaternary Reed-Muller code,and showed that this code has similar properties as a Reed-Muller code over afinite field. Many authors had theoretical interest in codes over residue rings,and more generally over Galois rings.Recently, Bhaintwal and Wasan [3] treated the Generalized Reed-Muller (GRM)1odes over Z p s for a prime power p s . Our purpose is to present a generalizationof GRM codes over Z p s to GRM codes over Galois rings.This paper is organized as follows. In section 2, we give a multivariate approachfor the GRM codes over Galois rings. In section 3, the standard generatormatrix for a GRM code over a Galois ring is given. In section 4, we prove thatthe images of the GRM codes over Galois rings under the projection map arethe usual GRM codes over finite fields. In section 5, the rank of a GRM codeover a Galois ring is given. The section 6 will examine the trace descriptionsof the Kerdock codes over Galois rings and the GRM codes over Galois rings.In section 7, the dual of a GRM code over a Galois ring is presented. And insection 8, the minimum distance of a GRM code over a Galois ring is provided. Throughout this paper, we consider the Galois ring GR ( p s , r ) of characteris-tic p s and cardinality p sr , where p is a prime number and s, r are integers ≥ . GR ( p s , r ) is a local ring with maximal ideal pGR ( p s , r ) and residue field GR ( p s , r ) /pGR ( p s , r ) = F q , where q = p r .Let h ( x ) ∈ GR ( p s , r )[ x ] be a monic basic primitive polynomial of degree m ≥ dividing x q m − − and having ξ as a root of order q m − in the Galoisring GR ( p s , r )[ x ] / ( h ( x )) = GR ( p s , rm ) . ξ is called a primitive element of GR ( p s , rm ) . Let n = q m − and T m = { , , ξ, ξ , . . . , ξ n − } . { , ξ, ξ , . . . , ξ m − } is a basis of the free module GR ( p s , rm ) of rank m over GR ( p s , r ) , and we have GR ( p s , rm ) = GR ( p s , r )[ ξ ] .We fix the notations R = GR ( p s , rm ) and L = GR ( p s , r ) .Each element ξ i ∈ T m can uniquely be expressed as ξ i = b i + b i ξ + b i ξ + . . . + b mi ξ m − , (1)where b ji ∈ L , ≤ i ≤ n − , ≤ j ≤ m .We adopt the convention ξ ∞ = 0 .Let b i = ( b i , b i , b i , . . . , b mi ) , ≤ i ≤ n − , and b ∞ = (0 , , . . . , .Let X be the set of variables x , x , . . . , x m and let P ( X ) be a polynomial inthese variables with coefficients from L . The degree of a nonzero monomial x i x i . . . x i m m is P mk =1 i k and the degree of a polynomial P ( X ) is the largestdegree of a monomial in P ( X ) . We define deg(0) = −∞ .We define the evaluation map ev : L [ X ] −→ L q m P ( X ) ( P ( b ∞ ) , P ( b ) , P ( b ) , . . . , P ( b n − )) (2)2et S = { P ( X ) ∈ L [ X ] | deg x i ( P ( X )) ≤ q − , ≤ i ≤ m } . Let ν be an integer such that ≤ ν ≤ m ( q − . Then the ν th order GeneralizedReed-Muller code of length q m over L is defined by RM L ( ν, m ) = { ev ( P ( X )) | P ( X ) ∈ S , deg( P ( X )) ≤ ν } . (3)The shortened Generalized Reed-Muller code of length q m − and order ν over L denoted by RM L ( ν, m ) − is the code obtained from RM L ( ν, m ) by puncturingat the first position. The component-wise product of any two elements u = ( u , u , . . . , u n ) and v = ( v , v , . . . , v n ) of L n +1 is defined by uv = ( u v , u v , . . . , u n v n ) . (4)By (1), let us consider the ( m + 1) × q m matrix G := (cid:18) ...
10 1 ξ ξ ... ξ n − (cid:19) .G can be expressed as G := ... b b b ... b n − b b b ... b n − ... ... ... ... ... ... b m b m b m ... b mn − . (5)The ith row of G is denoted by v i , ≤ i ≤ m . Thus, the v i are q m -tuplesover L . In particular, v is the all one tuple q m . Let ν be an integer such that ≤ ν ≤ m ( q − .From section 2, each P ( X ) ∈ S can be expressed as P ( X ) = X ≤ i j ≤ q − a i ,...,i m x i x i . . . x i m m , where a i ,...,i m ∈ GR ( p s , r ) .By (2),(4) and (5), we have ev ( x i x i . . . x i m m ) = v i v i . . . v i m m . And since the map ev is linear, we have ev ( P ( X )) = X ≤ i j ≤ q − a i ,...,i m v i v i . . . v i m m ν th order Generalized Reed-Muller code RM L ( ν, m ) of length q m over L is defined to be the code generated by all tuples of the form v i v i . . . v i m m , ≤ i j ≤ q − , ≤ j ≤ m , m X j =1 i j ≤ ν. (6) RM L (0 , m ) is the repetition code of length q m over L .Let G ν be the matrix whose rows consist of all tuples in (6). G ν is called thestandard generator matrix of RM L ( ν, m ) . The coordinates of any tulpe in G ν are numbered ∞ , , , . . . , n − . Recall that q = p r , n = q m − and L = GR ( p s , r ) . Since L/pL = F q , considerthe projection map which is defined by reduction modulo pα : L −→ F q a ¯ a (7)This map is extended to α : L [ x ] −→ F q [ x ] f ( x ) = X i a i x i ¯ f ( x ) = X i ¯ a i x i and α : L q m −→ ( F q ) q m v = ( a , a , . . . , a n ) ¯ v = (¯ a , ¯ a , . . . , ¯ a n ) (8) We have α ( RM L ( ν, m )) = RM F q ( ν, m ) where RM F q ( ν, m ) is the usual GRM code of order ν ( ≤ ν ≤ m ( q − ) and of length q m over thefinite field F q [6].Proof. By (8) and (6), we have α ( v i v i . . . v i m m ) = ¯ v i ¯ v i . . . ¯ v i m m and the q m -tuples ¯ v i ¯ v i . . . ¯ v i m m where ≤ i j ≤ q − , ≤ j ≤ m , P mj =1 i j ≤ ν form a basis for the GRM code RM F q ( ν, m ) . Hence the result. Consider the q m -tuples v i v i . . . v i m m , ≤ i j ≤ q − , ≤ j ≤ m. (9)4 .1 Proposition. The q m -tuples in (9) form a basis for the free L -module L q m where L = GR ( p s , r ) .Proof. Let ¯ v i be the image of v i in ( F q ) q m , ≤ i ≤ m . From the theory ofGRM codes over finite fields, we know that the vectors ¯ v i ¯ v i . . . ¯ v i m m , ≤ i j ≤ q − , ≤ j ≤ m form a basis for ( F q ) q m over F q .Let v ∈ L q m . Then ¯ v ∈ ( F q ) q m , and there exist constants a (0) i ,...,i m ∈ F q suchthat ¯ v = X ≤ i j ≤ q − a (0) i ,...,i m ¯ v i ¯ v i . . . ¯ v i m m . Then we have v = X ≤ i j ≤ q − a ′ (0) i ,...,i m v i v i . . . v i m m + p u for some u ∈ L q m and a ′ (0) i ,...,i m ∈ L .There exist constants a ′ (1) i ,...,i m ∈ L such that u = X ≤ i j ≤ q − a ′ (1) i ,...,i m v i v i . . . v i m m + p u for some u ∈ L q m .Continuing in this way and noting that p s = 0 in L , we get constants a ′ (2) i ,...,i m , . . . , a ′ ( s − i ,...,i m ∈ L such that v = X ≤ i j ≤ q − ( a ′ (0) i ,...,i m + pa ′ (1) i ,...,i m + . . . + p s − a ′ ( s − i ,...,i m ) v i v i . . . v i m m . Hence, each v ∈ L q m can be expressed as a linear combination of the tuples v i v i . . . v i m m , ≤ i j ≤ q − , ≤ j ≤ m . Since these tuples are q m innumber and L q m is a free module of rank q m over L , they must form a basis for L q m . Let m be a positive integer such that rm ≥ s and q = p r . Then,for ≤ ν ≤ m ( q − , the GRM code RM L ( ν, m ) is a free L -module of rank k ,where k = ν X i =0 m X j =0 ( − j (cid:18) mj (cid:19)(cid:18) i − jq + m − i − jq (cid:19) . Proof.
By (6), the elements of the set B = { v i v i . . . v i m m | ≤ i j ≤ q − , m X j =1 i j ≤ ν } (10)5pan the GRM code RM L ( ν, m ) . Since B is a subset of the set { v i v i . . . v i m m | ≤ i j ≤ q − } which forms a basis for the free L -module L q m , then B must belinearly independent. Thus, B is a basis for RM L ( ν, m ) and hence RM L ( ν, m ) isa free module over L . The images of the elements in B under the map α generatethe GRM code RM F q ( ν, m ) over F q . Also, these images are linearly independentover F q . Therefore, the elements ¯ v , where v ∈ B , form a basis for RM F q ( ν, m ) = α ( RM L ( ν, m )) , and we have Rank RM L ( ν, m ) = Rank RM F q ( ν, m ) . It is knownfrom the theory of GRM codes over finite fields that Rank RM F q ( ν, m ) = ν X i =0 m X j =0 ( − j (cid:18) mj (cid:19)(cid:18) i − jq + m − i − jq (cid:19) . Hence the result.
The rank of the GRM code RM L ( ν, m ) is just the number ofways we can place ν or fewer objects in m cells where no cell is to contain morethan q − objects. Each element c ∈ R = GR ( p s , rm ) has a unique p -adic representation c = ξ + pξ + p ξ + . . . + p s − ξ s − (11)where ξ , ξ , ξ , . . . , ξ s − ∈ T m = { , , ξ, ξ , . . . , ξ n − } . Under this representa-tion, the Frobenius automorphism is defined by f : R −→ R c = ξ + pξ + p ξ + . . . + p s − ξ s − c f = ξ q + pξ q + p ξ + . . . + p s − ξ qs − where q = p r . f is an automorphism of R , fixes only elements of L = GR ( p s , r ) ,and generates the group automorphism of R , which is cyclic of order m . Notethat when s = 1 , f is the usual Frobenius automorphism for F q m .The relative trace map is defined by T : R −→ Lc T ( c ) = c + c f + c f + . . . + c f m − .T is a linear transformation over L . Let m be a positive integer such that rm ≥ s and n = q m − with q = p r . Let h ( x ) ∈ L [ x ] be a monic basic primitive polynomial of degree m dividing x n − and having ξ as a root of order n in R . Let g ( x ) be the reciprocal polynomialof ( x n − / (( x − h ( x )) . The shortened Kerdock code K − is the cyclic code of6ength q m − over L = GR ( p s , r ) with generator polynomial g ( x ) .Since g ( x ) | x n − , K − is a free cyclic code of rank n − deg g ( x ) = m + 1 over L . A generator matrix of K − is G − := (cid:18) ... ξ ξ ... ξ n − (cid:19) . The Kerdock code K of length q m over L is obtained by adding an overall parity-check to K − .Since rm ≥ s and P n − i =0 ξ i = 0 , the zero-sum check for the first row of G − is and for the second row, it is . Thus, a generator matrix for K is G := (cid:18) ...
10 1 ξ ξ ... ξ n − (cid:19) where the elements in the second row of G are considered to be m -tuples over L . Thus we have K = RM L (1 , m ) . Let m be a positive integer such that rm ≥ s and n = q m − with q = p r . Let ξ be a primitive element of R = GR ( p s , rm ) . Then K − and K have the following trace descriptions over R K − = { ǫ n + v ( λ ) | ǫ ∈ L , λ ∈ R} , where n is the all one tuple of length n and v ( λ ) = ( T ( λ ) , T ( λξ ) , T ( λξ ) , . . . , T ( λξ n − )) , K = { ǫ n +1 + u ( λ ) | ǫ ∈ L , λ ∈ R} , where u ( λ ) = (0 , T ( λ ) , T ( λξ ) , T ( λξ ) , . . . , T ( λξ n − )) . Proof.
1. Let C = { ǫ n + v ( λ ) | ǫ ∈ L , λ ∈ R} . Let h ( x ) be the monic basicprimitive polynomial of degree m in L ( x ) dividing x n − such that h ( ξ ) = 0 .Let h ∗ ( x ) be the reciprocal polynomial of h ( x ) , i.e. h ∗ ( x ) = x m h ( x ) . Fromthe definition of K − , the check polynomial of K − is (1 − x ) h ∗ ( x ) . Clearly, − x annihilates the tuple ǫ n and h ∗ ( x ) annihilates v ( λ ) . Thus (1 − x ) h ∗ ( x ) annihilates C . It follows that C ⊆ K − . On the other hand, we have Card C =Card K − = ( p sr ) m +1 .2. K is a parity check extension of K − and the zero-sum check for ǫ n is ǫ andthe zero-sum check for v ( λ ) is . Thus, the result follows from 1. Let k be any integer such that ≤ k ≤ q m − , where q = p r . Then, k canuniquely be expressed as k = m − X i =0 a i q i , ≤ a i ≤ q − . The q -weight of k is defined by w q ( k ) = m − X i =0 a i . Rank RM L ( ν, m ) = Card( { j | ≤ j ≤ q m − , w q ( j ) ≤ ν } ) . Let m be a positive integer such that rm ≥ s and n = q m − with q = p r . Let ≤ ν ≤ m ( q − . Then, RM L ( ν, m ) is generated by therepetition code RM L (0 , m ) together with all q m -tuples of the form (0 , T ( λ j ) , T ( λ j ξ j ) , T ( λ j ξ j ) , . . . , T ( λ j ξ ( n − j )) (12) where j ranges over a system of representatives of those cyclotomic cosets mod-ulo q m − for which w q ( j ) ≤ ν and λ j ranges over R = GR ( p s , rm ) .Proof. Let C be the code generated by the repetition code RM L (0 , m ) togetherwith all tuples in (12). Since the matrix G in (5) is a generator matrix for theKerdock code K , then from Theorem 6.1, for each row v j , j = 1 , , . . . , m of G ,there exists a unique λ j ∈ R such that v j = (0 , T ( λ j ) , T ( λ j ξ j ) , T ( λ j ξ j ) , . . . , T ( λ j ξ ( n − j )) . Thus, the lth coordinate of a tuple v i v i . . . v i m m , ≤ i j ≤ q − , ≤ j ≤ m , m X j =1 i j ≤ ν in the standard generator matrix of the GRM code RM L ( ν, m ) is of the form T ( λ z ) i T ( λ z ) i . . . T ( λ m z ) i m , ≤ i j ≤ q − , m X i =1 i j ≤ ν, where z = ξ l with ξ ∞ = 0 . Now for a particular i and j , we have T ( λ i z ) T ( λ j z ) = m − X u =0 ( λ i z ) f u m − X v =0 ( λ j z ) f v = m − X u =0 λ f u i z q u m − X v =0 λ f v j z q v = T ( λ i λ j z ) + T ( λ i λ fj z q ) + T ( λ i λ f j z q ) + . . . + T ( λ i λ f m − j z q m − )= m − X k =0 T ( λ i λ f k j z q k ) = X t T ( µ t z t ) . where t = 1 + q k and µ t = λ i λ f k j , ≤ k ≤ m − . In general, if P mj =1 i j = a ≥ ,then T ( λ z ) i T ( λ z ) i . . . T ( λ m z ) i m = X t T ( µ t z t ) , t = 1 + q j + q j + . . . + q j a − , ≤ j k ≤ m − , ≤ k ≤ a − , and µ t isthe corresponding product of the powers of λ , λ , . . . , λ m .It is easy to see that in this expansion of any T ( λ z ) i T ( λ z ) i . . . T ( λ m z ) i m ,the corresponding powers t of z are some representatives of cyclotomic cosetsmodulo q m − and w q ( t ) ≤ P mk =1 i k . It follows that each tuple v i v i . . . v i m m inthe generator matrix of the GRM code RM L ( ν, m ) is a linear combination of theall one tuple q m and the tuples (0 , T ( λ j ) , T ( λ j ξ j ) , T ( λ j ξ j ) , . . . , T ( λ j ξ ( n − j )) ,where j ranges over a set of coset representatives modulo q m − with w q ( j ) ≤ P mk =1 i k ≤ ν and λ j ranges over R . Hence RM L ( ν, m ) ⊆ C .Conversely, let C − be the code obtained from C by puncturing at the first posi-tion. That is, C − is generated by the all one tuple n together with all tuplesof the form ( T ( λ j ) , T ( λ j ξ j ) , T ( λ j ξ j ) , . . . , T ( λ j ξ ( n − j )) , where j and λ j are asin (12).Since w q ( j ) ≤ ν , it is easy to verify that all these generators are annihilated bythe polynomial f ∗ ν ( x ) = (1 − x ) Y ≤ j ≤ qm − w q ( j ) ≤ ν (1 − ξ j x ) where f ∗ ν ( x ) is the reciprocal polynomial of f ν ( x ) = ( x − Y ≤ j ≤ qm − w q ( j ) ≤ ν ( x − ξ j ) = Y ≤ j ≤ qm − w q ( j ) ≤ ν ( x − ξ j ) Let g ν ( x ) be the reciprocal polynomial to the polynomial g ∗ ν ( x ) = x q m − − f ν ( x ) = Y ≤ j ≤ qm − w q ( j ) >ν ( x − ξ j ) and denote by C ν = ( g ν ( x )) the L -cyclic code generated by g ν ( x ) . Then, f ∗ ν ( x ) is the check polynomial of C ν . Therefore, C − ⊆ C ν . Thus, RM L ( ν, m ) − ⊆ C − ⊆C ν . Clearly, g ν ( x ) = Y ≤ j ≤ qm − w q ( j ) >ν (1 − ξ j x ) . We have
Rank C ν = q m − − deg g ν ( x )= Card( { j | ≤ j ≤ q m − , w q ( j ) ≤ ν } )= Rank RM L ( ν, m )= Rank RM L ( ν, m ) − . It follows that RM L ( ν, m ) − = C − = C ν . RM L ( ν, m ) − is a L -cyclic code generated by the polynomial g ν ( x ) = Y ≤ j ≤ qm − w q ( j ) ≤ m ( q − − ν − ( x − ξ j ) . (13)9 roof. By the proof of Theorem 6.3, we have RM L ( ν, m ) − = ( g ν ( x )) with g ν ( x ) = Y ≤ j ≤ qm − w q ( j ) >ν (1 − ξ j x ) . The zeros of g ν ( x ) are all ξ − j with w q ( j ) > ν . Since ξ − j = ξ q m − − j and w q ( q m − − j ) = m ( q − − w q ( j ) , we have g ν ( x ) = Y ≤ j ≤ qm − w q ( j ) >ν ( x − ξ − j ) = Y ≤ j ≤ qm − w q ( j ) >ν ( x − ξ q m − − j ) . Let J = q m − − j . We have J = 0 and w q ( J ) = m ( q − − w q ( j ) . Thus, w q ( j ) = m ( q − − w q ( J ) > ν . This implies that w q ( J ) < m ( q − − ν . Then g ν ( x ) = Y
Rank RM L ( ν, m ) + Rank RM L ( µ, m ) = q m .Thus, Rank RM L ( µ, m ) = Rank RM L ( ν, m ) ⊥ .And we have RM L ( µ, m ) = RM L ( ν, m ) ⊥ . The shortened GRM code RM L ( ν, m ) − is a subcode of a BCHcode of length q m − over L whose roots include ξ, ξ , . . . , ξ ( R +1) q Q − where ξ is a primitive element of R = GR ( p s , rm ) , and Q and R are the quotient andremainder respectively, resulting from dividing µ + 1 = m ( q − − ν by q − .Proof. Let d be the smallest integer such that w q ( d ) = m ( q − − ν = ( q − Q + R , ≤ R < q − . Therefore, we must have d = Rq Q + ( q − q Q − +( q − q Q − + . . . + ( q − q + ( q −
1) = ( R + 1) q Q − .Also, every integer less than d has q -weight less than or equal to m ( q − − ν − . It follows from (13) that all elements ξ, ξ , . . . , ξ ( R +1) q Q − are roots of RM L ( ν, m ) − . Thus, RM L ( ν, m ) − is a subcode of a primitive BCH code oflength q m − over L .Consequently, from BCH bound on codes over Galois rings [2], RM L ( ν, m ) − has minimum distance at least ( R + 1) q Q − , where Q and R are the quotientand remainder respectively, resulting from dividing µ + 1 = m ( q − − ν by q − . It can be easily seen from the structure of RM L ( ν, m ) that the minimumdistance of RM L ( ν, m ) is equal to the minimum distance of RM L ( ν, m ) − . Hencethe minimum distance of RM L ( ν, m ) is at least ( R + 1) q Q − . The GRM code RM L ( ν, m ) has minimum distance ( R + 1) q Q − , where Q and R are the quotient and remainder respectively, resulting fromdividing µ + 1 = m ( q − − ν by q − .Proof. since the minimum distance of RM L ( ν, m ) is at least ( R + 1) q Q − , weonly need to show a tuple of weight ( R + 1) q Q − in RM L ( ν, m ) . The image α ( RM L ( ν, m )) = RM F q ( ν, m ) has minimum distance exactly ( R + 1) q Q − . Let u = ( u ∞ , u , u , . . . , u n − ) be a vector of weight ( R + 1) q Q − in RM F q ( ν, m ) .Let I = Supp( u ) = { i | u i = 0 } the support of u . Then, Card( I ) = ( R +1) q Q − .Then, there exists a vector v = ( v ∞ , v , v , . . . , v n − ) ∈ RM L ( ν, m ) such that12 ( v ) = ¯ v = u i.e. (¯ v ∞ , ¯ v , ¯ v , . . . , ¯ v n − ) = ( u ∞ , u , u , . . . , u n − ) . Thus, ¯ v i = u i for all i .If i / ∈ I , then u i = 0 . Thus, v i is in pL , and p s − v i = 0 .If i ∈ I , then u i = 0 and v i / ∈ pL , i.e. v i is an invertible element of L , and p s − v i = 0 . Therefore, p s − v ∈ RM L ( ν, m ) and p s − v is of weight ( R +1) q Q − .Hence, RM L ( ν, m ) has minimum distance ( R + 1) q Q − . References [1] E. F. Assmus and J. D. Key,
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