[Galois-disjoint] Polycyclic codes over finite chain rings LAWCI - PowerPoint PPT Presentation

[Galois-disjoint] Polycyclic codes over finite chain rings LAWCI 2018, Campinas Thomas Blackford, Alexandre Fotue Tabue, Edgar Mart´ ınez-Moro July 25th, 2018

Finite Commutative Chain Rings A finite ring S is a finite chain ring (FCR) if S is local and principal. If θ is a generator of the maximal ideal J ( S ) of S the ideals of S form a chain as follows: { 0 S } = J ( S ) s � J ( S ) s − 1 � · · · · · · � J ( S ) � S , and J ( S ) t = θ t S for 0 ≤ t < s . Any FCR is the Galois extension of Z p a [ θ ] of degree r , for some root θ of an Eisenstein polynomial over Z p a of degree e satisfying θ s − 1 � = θ s = 0 S . Whence S r isomorphic to GR ( p a , r )[ θ ] . We will call ( p a , r , e , s ) the parameters of the FCR. 2/21

Example Consider Z 4 [ ζ ][ x ] S = � x 2 + 2 , 2 x � , where ζ is a root of the basic primitive polynomial f ( X ) = X 2 + X + 1 ∈ Z 4 [ X ]. R e has 64 elements, and is additively equivalent to Z 2 ⊕ Z 2 ⊕ Z 4 ⊕ Z 4 . Z 4 [ x ] Moreover, it is a (Galois) extension of the ring R = � x 2 +2 , 2 x � with S = R [ ζ ], and also S / � x � ∼ = F 4 = F 2 ( ζ ). Similarly, Z 4 [ ζ ] is a Galois ring as an extension of S = Z 4 . 3/21

Tree Z 4 [ ζ ][ x ] � x 2 +2 , 2 x � 3 2 2 Z 4 [ ζ ] Z 4 [ x ] � x 2 +2 , 2 x � 2 3 2 Z 4 4/21

Codes over a FCR A S -linear code of length n is a submodule of the S -module S n . An S -linear code over S is free, if it is free as an S -module. A matrix G is called a generator matrix for an S -linear code C , if the rows of G span C and none of them can be written as an S -linear combination of the other rows of G . 5/21

Codes over a FCR A matrix G is in the standard form if it is of the form  · · ·  I k 0 G 0 , 1 G 0 , 2 G 0 , s − 1 G 0 , s 0 θ I k 1 θ G 1 , 2 · · · θ G 1 , s − 1 θ G 1 , s   G =  U ,   · · · · · · · · · · · · · · · · · ·  θ s − 1 I k s − 1 θ s − 1 G s − 1 , s 0 0 0 · · · where I k t is an identity matrix of order k t (for 0 ≤ t < s ) and U is a suitable permutation matrix. The s -tuple ( k 0 , k 1 , · · · , k s − 1 ) is called type of G . 6/21

Polycyclic codes over a FCR Consider the n × n -matrices     0 b 0 · · · b n − 2 b n − 1 . 0 .     . I n − 1 D a :=  and E b :=    .  .     0 . I n − 1    · · · 0 a 0 a 1 a n − 1 with associate vectors a := ( a 0 , a 1 , · · · , a n − 1 ) and b := ( b 0 , b 1 , · · · , b n − 1 ) of S n , respectively. 7/21

Polycyclic codes over a FCR an S -linear code of length n is right polycyclic with associate vector a (or simply, right a -cyclic), if it is invariant by right multiplication of the matrix D a . It is left b -cyclic if it is invariant by right multiplication of the matrix E b . Any right a -cyclic with a 0 ∈ S × is also left b -cyclic where b j = − a j +1 a − 1 for j < n − 1 and b n − 1 = a − 1 0 . Henceforth, we assume that 0 the right a -cyclic codes will satisfy a ∈ S × × S n − 1 , and we will simply write a -cyclic codes. 8/21

Polycyclic codes over a FCR Consider the S -module monomorphism S n Ψ : ֒ → S [ X ] c 0 + c 1 X + · · · + c n − 1 X n − 1 . . ( c 0 , c 1 , · · · , c n − 1 ) �→ C is an a -cyclic code if and only if Ψ( C ) is an ideal in S [ X ] contained the ideal � X n − Ψ( a ) � in S [ X ] generated by X n − Ψ( a ) . the a -period of an a -cyclic over S will be i ∈ N \{ 0 } : X n − π (Ψ( a )) divides X i − 1 � � ℓ a := min . where π is the natural projection S → S / J ( S ). The quotient ring S [ X ] / � X n − Ψ( a ) � is a principal ideal ring, if either S is a field or X n − π (Ψ( a )) is square free. 9/21

Free codes From now on ( ℓ a , p ) = 1 A free polycyclic code C = P ( S ; n ; g ) over S is defined as follows: Consider g = g 0 + g 1 X + · · · + g k − 1 X k − 1 + X k over S and C is gen- erated as the spand of its cyclic shifts (as n -vectors). The polynomial g is called the generator polynomial of C . If g 0 ∈ S × , and there is an a ∈ ( S × ) × S n − 1 such that g divides X n − Ψ( a ) Then � � c ∈ S n : g divides Ψ( c ) P ( S ; n ; g ) = and P ( S ; n ; g ) is a -cyclic. 10/21

Non-free codes Every non-free, non-zero polycyclic code C over S admits a generator set [strong Gr¨ obner basis] of the form � � θ λ 1 g 1 ; θ λ 2 g 2 ; · · · ; θ λ u g u ∈ S [ X ] such that � θ λ 1 g 1 ; θ λ 2 g 2 ; · · · ; θ λ u g u � Ψ( C ) = , and 1. 0 ≤ λ 1 < λ 2 < · · · < λ u < s ; 2. for 1 ≤ i ≤ u , g i is monic; 3. n > deg ( g 1 ) > deg ( g 2 ) > · · · > deg ( g u ) > 0; 4. for 0 ≤ i ≤ u , θ λ i +1 g i ∈ � θ λ i +1 g i +1 ; θ λ i +2 g i +2 ; · · · ; θ λ u g u � . 11/21

Non-free codes A strong Gr¨ obner basis is not necessarily unique. However, u the cardinality of the basis, k 1 , k 2 , · · · , k u the degrees of its polynomials g 1 , g 2 , · · · , g u and the exponents λ 1 , λ 2 , · · · , λ u are uniquely deter- mined. u θ λ i ( P ( S ; k i − 1 ; g i )) � C = i =1 12/21

Euclidean orthogonality Any S -linear code C of length n is a -cyclic, if and only if C ⊥ is a -sequential, i.e. ( c 1 , · · · , c n − 1 , � c , a � ) ∈ C , for all c ∈ C . Sequencial codes are identified with ideals in constacyclic ambient spaces S [ X ] / � X n − Ψ( a ) � . 13/21

Galois action Let σ be a generator of Aut S r ( S ) . This generator σ naturally induces an automorphism of S [ X ] and S n as follows: � σ ( f i ) X i , σ ( f ) = i where σ ( c ) = ( σ ( c 0 ) , σ ( c 1 ) , · · · , σ ( c n − 1 )) . Let C be an S -linear code 1. The � σ r � - image of C is defined as: σ r ( C ) := { σ r ( c ) : c ∈ C } . 2. The code C is � σ r � - disjoint if σ ir ( C ) ∩ C = { 0 } for all 1 ≤ i < d . 3. The code C is completed � σ r � - disjoint if S n = C + σ r ( C ) + σ 2 r ( C ) + · · · + σ r ( d − 1) ( C ) . 14/21

Galois action Let C be a non-free, non-zero a -cyclic code over S with strong Gr¨ obner bases � � θ λ 1 g 1 ; θ λ 2 g 2 ; · · · ; θ λ u g u . Then σ i ( C ) is σ i ( a )-cyclic, and � θ λ 1 σ i ( g 1 ) ; θ λ 2 σ i ( g 2 ); · · · ; θ λ u σ i ( g u ) � obner basis for σ i ( C ) , for all 0 ≤ i < m . is a Gr¨ 15/21

Galois action Let C := P ( S ; n ; g ) with generator polynomial g . Then C is � σ r � - disjoint if, and only if deg ( µ ( g , σ ir ( g ))) ≥ n , for 1 ≤ i ≤ d − 1 . Let C be an S -linear code over S of length n . If C is completed � σ r � -disjoint then C is free. 16/21

Trace codes and restrictions Let C be an S -linear code of length n . The restriction code Res r ( C ) of C to S r is defined to be Res r ( C ) := C ∩ ( S r ) n , and the trace code Tr d ( C ) of C to S r is defined as: � � Tr d ( C ) := ( Tr d ( c 0 ) , Tr d ( c 1 ) , · · · , Tr d ( c n − 1 )) : c ∈ C , d − 1 σ ir . � where Tr d = i =0 17/21

Delsarte Let ρ ∈ S [ X ] . Denote by { ρ } a the unique polynomial such that X n − Ψ( a ) divides ρ − { ρ } a and deg ( { ρ } a ) < n . One defines the inner- product by � x ; y � a := ( { Ψ( x )Ψ( y ) } a ) (0 S ) . If a ∈ ( S × ) × S n − 1 , then the bilinear form ( x ; y ) �→ � x ; y � a is non degenerate. The annihilator of an a -cyclic code is Ann S ( C ) := { y ∈ S n : � y ; c � a = 0 S for all c ∈ C �} . 18/21

Delsarte Ann S ( C ) is also an a -cyclic code, C ⊥ and Ann S ( C ) have the same type, Ann S ( Ann S ( C )) = C and | Ann S ( C ) | × | C | = | S | n . Let C be a non-free, non-zero a -cyclic code over S with strong Gr¨ obner bases � � θ λ 1 g 1 ; θ λ 2 g 2 ; · · · ; θ λ u g u . Let λ u +1 = s and g 0 = f . For i ∈ { 1; 2; · · · ; u +1 } , λ ∗ i := s − λ u − i +2 and h i a monic polynomial over S such that π ( h i g u − i +1 ) := π ( f ) . Then Ann S ( C ) is an a -cyclic code with strong Gr¨ obner basis � � θ λ ∗ 1 h 1 ; θ λ ∗ 2 h 2 ; · · · ; θ λ ∗ u +1 h u +1 19/21

Delsarte Let C be an a -cyclic code over S . Then Tr d ( Ann S ( C )) = Ann S r ( Res r ( C )) . 20/21

Thank you for your attention! 21/21