New descriptions of the weighted Reed-Muller codes and the homogeneous Reed-Muller codes
aa r X i v : . [ c s . I T ] J a n New descriptions of the weighted Reed-Mullercodes and the homogeneous Reed-Muller codes
Harinaivo ANDRIATAHINYMention: Mathematics and Computer Science,Domain: Sciences and Technologies,University of Antananarivo, Madagascare-mail: [email protected] Harinoro RAKOTOMALALAMention: Meteorology,Domain: Sciences of the Engineer,Higher Polytechnic School of Antananarivo (ESPA),University of Antananarivo, Madagascare-mail: [email protected] 15, 2018
Abstract
We give a description of the weighted Reed-Muller codes over a primefield in a modular algebra. A description of the homogeneous Reed-Mullercodes in the same ambient space is presented for the binary case. Adecoding procedure using the Landrock-Manz method is developed.
Keywords: weighted Reed-Muller codes, homogeneous Reed-Muller codes, mod-ular algebra, Jennings basis, decoding.MSC 2010: 94B05, 94B35, 12E05.
It is well known that the Generalized Reed-Muller (GRM) codes of length p m over the prime field F p can be viewed as the radical powers of the modular alge-bra A = F p [ X , . . . , X m − ] / ( X p − , . . . , X pm − − ([1],[4],[5]). A is isomorphicto the group algebra F p [ F p m ] .The weighted Reed-Muller codes and the homogeneous Reed-Muller codes areclasses of codes in the Reed-Muller family. The Jennings basis are used to de-scribe the GRM codes over F p . We utilize the elements of the Jennings basis forthe description of the weighted Reed-Muller codes and the homogeneous Reed-Muller codes in A . P. Landrock and O. Manz developed a decoding algorithm1or the binary Reed-Muller codes in [9]. We use here the same method for thebinary homogeneous Reed-Muller codes.The weighted Reed-Muller codes can be considered as a generalization of theGRM codes. Some classes of the weighted Reed-Muller codes are algebraic-geometric codes.The homogeneous Reed-Muller codes are subcodes of the GRM codes. In gen-eral, they have a much better minimum distance than the GRM codes.We give, in section 2, the definition and some properties of the weighted Reed-Muller codes. We consider here the affine case. In section 3, a description ofthe weighted Reed-Muller codes over F p in the quotient ring A is presented.In section 4, we recall the definition and the parameters of the homogeneousReed-Muller codes. In section 5, we describe the homogeneous Reed-Mullercodes over the two elements field F in A (with p = 2 ). In section 6, we use theLandrock-Manz method to construct a decoding procedure for the homogeneousReed-Muller codes in the binary case. In section 7, an example is given. The definition and the properties of the weighted Reed-Muller codes presentedin this section are from [11]. Let F q the field of q = p r elements where p isa prime number and r ≥ is an integer. Let ( F q ) m be the m -dimensionalaffine space defined over F q . F q [ Y , Y , . . . , Y m − ] is the ring of polynomials in m variables with coefficients in F q . If we attach to each variables Y i a naturalnumber w i , called weight of Y i , we speak about the ring of weighted polynomials, W F q [ Y , Y , . . . , Y m − ] . The weighted degree of F ∈ W F q [ Y , Y , . . . , Y m − ] , isdefined as deg ̟ ( F ) = deg ̟ ( F ( Y , . . . , Y m − )) = deg( F ( Y w , . . . , Y w m − m − )) , where deg is the usual degree.We will, without loss of generality, always assume that the weights are ordered w ≤ w ≤ . . . ≤ w m − . Consider the evaluation map φ : W F q [ Y , . . . , Y m − ] −→ ( F q ) q m F φ ( F ) = ( F ( P ) , . . . , F ( P n )) (1)where P , . . . , P n ( n = q m ) is an arbitrary ordering of the elements of ( F q ) m .For w = w = . . . = w m − = 1 we have the following definition.The Generalized Reed-Muller codes of order ν ( ≤ ν ≤ m ( q − ) and length n = q m is defined by C ν ( m, q ) = φ ( V ( ν )) , where V ( ν ) = { F ∈ F q [ Y , . . . , Y m − ] | deg( F ) ≤ ν } . Let ω be a natural number and { w , . . . , w m − } be weights corresponding to thering of weighted polynomials W F q [ Y , . . . , Y m − ] . The weighted Reed-Muller2odes W RM C ω ( m, q ) of weighted order ω and length n = q m , corresponding tothe weights { w , . . . , w m − } is defined by W RM C ω ( m, q ) = φ ( V ̟ ( ω )) , (2)where V ̟ ( ω ) = { F ∈ W F q [ Y , . . . , Y m − ] | deg ̟ ( F ) ≤ ω } . (3)For a polynomial F ∈ F q [ Y , . . . , Y m − ] , F denotes the reduced form of F , i.e.the polynomial of lowest degree equivalent to F modulo the ideal ( Y qi − Y i , i =0 , . . . m − . For any subset M of F q [ Y , . . . , Y m − ] , the set M denotes the setof reduced elements of M . For F ∈ F q [ Y , . . . , Y m − ] , we have1. for every P ∈ ( F q ) m : F ( P ) = F ( P ) .2. if F ( P ) = 0 for all P ∈ ( F q ) m , then F = 0 .Given natural numbers ω, ν , and a set of weights { w , . . . , w m − } such that ≤ ν ≤ m ( q − and ≤ ω ≤ ( q − P mi =1 w i .Let ν max ( ω ) = Q ′ ( q −
1) + R ′ where Q ′ = max { Q | ω ≥ Q X i =0 ( q − w i } and R ′ = max { R | ω ≥ Q ′ X i =0 ( q − w i + Rw Q ′ +1 } . Given a natural number ω and a set of ordered weights { w , . . . , w m − } such that ω ≤ ( q − P m − i =0 w i . The code W RM C ω ( m, q ) isan F q -linear [ q m , k, d ] code with k = card( { ( e , . . . , e m − ) | m − X i =0 w i e i ≤ ω, ≤ e i < q } ) and d = q m − Q − ( q − R ) where Q and R are given by ν max ( ω ) = Q ( q −
1) + R, with ≤ R < q − . The set of monomials { m − Y i =0 Y e i i | m − X i =0 w i e i ≤ ω, ≤ e i < q } is a basis of V ̟ ( ω ) . 3 Description of the weighted Reed-Muller codesin A Consider the modular algebra A = F p [ X , X , . . . , X m − ] / ( X p − , . . . , X pm − − and the ideal I = (cid:0) X p − , . . . , X pm − − (cid:1) of the polynomial ring F p [ X , . . . , X m − ] , where F p is the prime field of p (aprime number) elements.Set x = X + I, . . . , x m − = X m − + I . Let us fix an order on the set ofmonomials n x i . . . x i m − m − | ≤ i , . . . , i m − ≤ p − o . Then A = p − X i =0 · · · p − X i m − =0 a i ...i m − x i . . . x i m − m − | a i ...i m − ∈ F p . (4)And we have the following identification: A ∋ P p − i =0 · · · P p − i m − =0 a i ...i m − x i . . . x i m − m − ←→ ( a i ...i m − ) ≤ i ,...,i m − ≤ p − ∈ ( F p ) p m .Hence the modular algebra A is identified with ( F p ) p m . P ( m, p ) denotes the vector space of the reduced polynomials in m variables over F p : P ( Y , . . . , Y m − ) = p − X i =0 · · · p − X i m − =0 u i ...i m − Y i . . . Y i m − m − | u i ...i m − ∈ F p . Consider a set of weights { w , . . . , w m − } and let ω be an integer such that ≤ ω ≤ ( p − w + . . . + w m − ) .When considering P ( m, p ) and A as vector spaces over F p , we have the followingisomorphism: ψ : P ( m, p ) −→ AP ( Y , . . . , Y m − ) p − X i =0 · · · p − X i m − =0 P ( i , . . . , i m − ) x i . . . x i m − m − (5)The set B := (cid:8) ( x − i . . . ( x m − − i m − | ≤ i , . . . , i m − ≤ p − (cid:9) (6)4s called the Jennings basis of A .Set [0 , p m −
1] = { , , , . . . , p m − } .Let i ∈ [0 , p m − . Consider its p -adic expansion i = m − X k =0 i k p k with ≤ i k ≤ p − for all k = 0 , . . . , m − .We need the following notations and definitions: i := ( i , . . . , i m − ) ,the p -weight of i is defined by wt p ( i ) := P m − k =0 i k ,and the p -weight of i with respect to the set of weights { w , . . . , w m − } is definedby W wt p ( i ) := m − X k =0 i k w k . (7) j ≤ i if j l ≤ i l for all l = 0 , , . . . , m − where j := ( j , . . . , j m − ) ∈ ([0 , p − m , x := ( x , . . . , x m − ) , x i := x i . . . x i m − m − .Consider the polynomial B i ( x ) := ( x − i . . . ( x m − − i m − ∈ A. (8)The following proposition is from [1]. We have H i ( Y ) = ψ − ( B i ( x )) , where ψ is the isomorphismdefined in (5), i.e. B i ( x ) = X j ≤ i H i ( j ) x j where H i ( Y ) := m − Y l =0 H i l ( Y l ) and H i ( Y ) = α i p − − i Y j =1 ( Y + j ) , with α i = − i ! mod p . We have deg ̟ ( H i ( Y )) = ( p − m − X l =0 w l − W wt p ( i ) . We now present a description of the weighted Reed-Muller code
W RM C ω ( m, p ) in the algebra A . 5 .3 Theorem. Consider a set of weights { w , . . . , w m − } and let ω be an integersuch that ≤ ω ≤ ( p − P m − l =0 w l . Then, the set B ω := { ( x − i . . . ( x m − − i m − | ≤ i k ≤ p − , m − X k =0 w k i k ≥ ( p − m − X k =0 w k − ω } forms a linear basis of the weighted Reed-Muller code W RM C ω ( m, p ) over F p in A .Proof. It is clear that B ω is a set of linearly independant elements because B ω ⊆ B .Let B i ( x ) := ( x − i . . . ( x m − − i m − ∈ B ω , i.e. ≤ i k ≤ p − , for all k = 0 , . . . , m − , and P m − k =0 w k i k ≥ ( p − P m − k =0 w k − ω .By the Proposition 3.1 and the Corollary 3.2, we have B i ( x ) = P j ≤ i H i ( j ) x j with H i ( Y ) = Q m − l =0 H i l ( Y l ) , H i ( Y ) = α i Q p − − ij =1 ( Y + j ) , and α i = − i ! mod p .We have deg ̟ ( H i ( Y )) = ( p − P m − l =0 w l − W wt p ( i ) ≤ ω .Thus H i ( Y ) ∈ V ̟ ( ω ) .Therefore, B i ( x ) ∈ W RM C ω ( m, p ) .It is clear that dim F p ( W RM C ω ( m, p )) = card( { i ∈ [0 , p m − | W wt p ( i ) ≤ ω } ) .On the other hand, we have card( B ω ) = card( { i ∈ [0 , p m − | W wt p ( i ) ≥ ( p − P m − k =0 w k − ω } ) .Consider the bijection θ : [0 , p m − −→ [0 , p m − i = m − X k =0 i k p k θ ( i ) = m − X k =0 ( p − − i k ) p k . We have
W wt p ( θ ( i )) = P m − k =0 w k ( p − − i k ) = ( p − P m − k =0 w k − W wt p ( i ) ,i.e. W wt p ( i ) = ( p − P m − k =0 w k − W wt p ( θ ( i )) .Thus, we have W wt p ( i ) ≤ ω ⇐⇒ W wt p ( θ ( i )) ≥ ( p − P m − k =0 w k − ω .Hence, card( { i ∈ [0 , p m − | W wt p ( i ) ≤ ω } ) = card( { i ∈ [0 , p m − | W wt p ( i ) ≥ ( p − P m − k =0 w k − ω } ) .The following Corollary is the famous result of Berman-Charpin ([1],[4],[5]). Consider the weights w = . . . = w m − = 1 and an integer ω such that ≤ ω ≤ m ( p − . Then, the set B ω := { ( x − i . . . ( x m − − i m − | ≤ i k ≤ p − , m − X k =0 i k ≥ m ( p − − ω } forms a linear basis of the GRM code C ω ( m, p ) = P m ( p − − ω over F p , where P is the radical power of A . The homogeneous Reed-Muller codes
In this section, we recall the definition and some properties of the homogeneousReed-Muller codes [3],[10]. F q denote the field of q = p r elements with p a primenumber and r ≥ an integer. For n = q m − , let { , P , . . . , P n } be the set ofpoints in ( F q ) m ordered in a fixed order.Let F q [ Y , . . . , Y m − ] d be the vector space of homogeneous polynomials in m variables over F q of degree d .Now d denote an integer such that ≤ d ≤ m ( q − . The d th order homogeneousReed-Muller (HRM) codes of length q m over F q is defined as HRM C d ( m, q ) := { ( F (0) , F ( P ) , . . . , F ( P n )) | F ∈ F q [ Y , . . . , Y m − ] d } . (9)Thus HRM C d ( m, q ) is a proper subcode of the GRM code C d ( m, q ) .The following theorem can be found in [3]. Let d such that ≤ d ≤ ( m − q − . The HRM code HRM C d ( m, q ) is an [ n + 1 , k, δ ] linear code with n + 1 = q m , k = X t ≡ dmod ( q − , 1) + s and ≤ s < q − . A First, we recall some results in the Proposition 3.1 for the special case p = 2 .In this section, we consider the ambiant space A = F [ X , . . . , X m − ] / ( X − , . . . , X m − − . We have B i ( x ) = ( x − i . . . ( x m − − i m − = X j ≤ i H i ( j ) x j where ≤ i k ≤ , for all k , H i ( Y ) := m − Y l =0 H i l ( Y l ) and H i ( Y ) = α i − i Y j =1 ( Y + j ) , α i = − i ! mod .Note that B (1 , ,..., ( x ) = ( x − . . . ( x m − − = ˆ1 is the "all one" word.Let d be an integer such that ≤ d ≤ m . The d th order homogeneous Reed-Muller (HRM) codes of length m over F is defined as HRM C d ( m, 2) := { ( F (0) , F ( P ) , . . . , F ( P n )) | F ∈ F [ Y , . . . , Y m − ] d } , where n = 2 m − . We now give the description of the binary HRM code HRM C d ( m, in A . Let d be an integer such that ≤ d ≤ m . The set { ( x − i . . . ( x m − − i m − + ˆ1 | ≤ i k ≤ , m > m − X k =0 i k ≥ m − d } forms a linear basis for the binary HRM code HRM C d ( m, .Proof. Let d such that ≤ d ≤ m .Consider the element B i ( x ) + ˆ1 = ( x − i . . . ( x m − − i m − + ˆ1 , where ≤ i k ≤ for all k and m > P m − k =0 i k ≥ m − d .Set D ( i ) := { j ∈ ( { , } ) m | j ≤ i } and C ( i ) := ( { , } ) m − D ( i ) .We have H i ( j ) = 0 for j ∈ C ( i ) .Thus B i ( x ) = X j ≤ i H i ( j ) x j = X j ∈ ( { , } ) m H i ( j ) x j . We have H i ( Y ) := m − Y l =0 H i l ( Y l ) , where H ( Y ) = 1 , H ( Y ) = Y + 1 . (10)Since P m − k =0 i k ≥ m − d , then B i ( x ) ∈ P m − d where P is the radical of themodular algebra A .And since P m − d = C d ( m, , then H i ( Y ) ∈ P d ( m, where P d ( m, is a linearspace generated by the set { Y i . . . Y i m − m − | ≤ i k ≤ , ≤ m − X k =0 i k ≤ d } . (11)We have B i ( x ) + ˆ1 = X j ∈ ( { , } ) m ( H i ( j ) + 1) x j . 8y (10) and (11), we have H i ( Y ) + 1 ∈ F [ Y , . . . , Y m − ] d .Note that F [ Y , . . . , Y m − ] d is a linear space generated by the set S := { Y i . . . Y i m − m − | ≤ i k ≤ , < m − X k =0 i k ≤ d } . Thus B i ( x ) + ˆ1 ∈ HRM C d ( m, .Note also that P m − k =0 i k = m if and only if i k = 1 for all k = 0 , . . . , m − .Set R := { ( x − i . . . ( x m − − i m − + ˆ1 | ≤ i k ≤ , m > P m − k =0 i k ≥ m − d } .We will show that dim F ( HRM C d ( m, R ) .We have dim F ( HRM C d ( m, F ( F [ Y , . . . , Y m − ] d ) = card( S ) .Consider the bijection β : ( { , } ) m −→ ( { , } ) m ( i , . . . , i m − ) (1 − i , . . . , − i m − ) Set R ′ := { i = ( i , . . . , i m − ) ∈ ( { , } ) m | P m − k =0 i k ≥ m − d } and S ′ := { i = ( i , . . . , i m − ) ∈ ( { , } ) m | P m − k =0 i k ≤ d } .It is clear that S ′ = β ( R ′ ) . Thus card( R ′ ) = card( S ′ ) .Since card( R ) = card( R ′ ) − and card( S ) = card( S ′ ) − , then card( R ) =card( S ) . In this section, we will follow Landrock-Manz as in [9].Let d be an integer such that ≤ d ≤ m . The HRM code HRM C d ( m, is oftype h m , P dt =1 (cid:0) mt (cid:1) , m − d i over F .Set b ( { i , . . . , i t } ) := ( x i − . . . ( x i t − ,where { i , . . . , i t } ⊆ { , , . . . , m − } . B m − d := { b ( η ) + ˆ1 | η ⊆ { , , . . . , m − } , m > card( η ) ≥ m − d } is a linearbasis of HRM C d ( m, .General results of the following Proposition can be found in [2]. We have1. b ( {} ) = 1 .2. b ( η ) .b ( κ ) = ( if η ∩ κ = {} ,b ( η ∪ κ ) otherwise . 3. The weight of the codeword w ( b ( { i , . . . , i t } )) = 2 t .4. b ( { , , . . . , m − } ) = ˆ1 the "all one" word.5. ˆ1 .b ( { η } ) = 0 if η = {} . η c := { , , . . . , m − } − η .Let c ∈ HRM C d ( m, be a transmitted codeword and v ∈ A the receivedvector, where A = F [ X , . . . , X m − ] / ( X − , . . . , X m − − Since HRM C d ( m, is (2 m − d − − -error correcting, we write v = c + f with w ( f ) ≤ m − d − − .We have c = X m − d ≤ card( η ) Consider the binary HRM code HRM C (5 , . This code is of type [32 , , .Set f E l = Q k = l ( x k − 1) = b ( { l } c ) , where l ∈ { , , , , } .The set { f E l + ˆ1 | l = 0 , , , , } forms a linear basis for HRM C (5 , .Let c = P i =0 τ i . ( f E i + ˆ1) be a transmitted codeword with τ i ∈ F and v ∈ A = F [ X , X , . . . , X ] / ( X − , . . . , X − the received vector. Since HRM C (5 , is -error correcting, we write v = c + f with w ( f ) ≤ .We have v. ( x j − 1) = ( c + f ) . ( x j − 1) = c. ( x j − 1) + f. ( x j − 1) = τ j . ˆ1 + f. ( x j − for j = 0 , , , , .Since w ( f. ( x j − ≤ w ( f ) .w (( x j − ≤ ∗ < , then we have thefollowing Proposition τ j = 0 if and only if w ( v. ( x j − < By multiplying v with ( x j − for j = 0 , , , , and utilizing the Proposition7.1, we obtain the coefficients τ , τ , τ , τ , τ of c . References [1] H. Andriatahiny, The Generalized Reed-Muller codes and the Radical Pow-ers of a modular algebra , British Journal of Mathematics and ComputerScience, 18(5):1-14, 2016.[2] E. F. Assmus and J. D. 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