Separability Test and Cyclic Convolutional Codes.J. Algebra, 123 (1989) P. Piret. Structure and...

Separability Test and Cyclic Convolutional Codes.1J. Gómez-Torrecillas ?, F. J. Lobillo ? and G. Navarro ‡

?Department of Algebra and CITIC, University of Granada‡Dep. of Computer Sciences and AI, and CITIC, University of Granada

XV Encuentro de Álgebra Computacional y Aplicaciones, Logroño, June 22th, 2016

1Research supported by grants MTM2013-41992-P and TIN2013-41990-R from Ministerio de Economía y Competitividad of the SpanishGovernment and from FEDERJ. Gómez-Torrecillas (UGR) CCC & Separable Ring Extensions. EACA2016 (1)

ReferencesS. Estrada, J.R. García-Rozas, J. Peralta and E.Sánchez-García, Group convolutional codes, Adv.Math. Commun. 2 (2008).H. Gluesing-Luerssen and W. Schmale. On cyclicconvolutional codes. Acta ApplicandaeMathematica, 82 (2004).K. Hirata and K. Sugano. On semisimpleextensions and separable extensions over noncommutative rings. Journal of the MathematicalSociety of Japan, 18 (1966).S. R. López-Permouth and S. Szabo. Convolutionalcodes with additional algebraic structure. J. PureAppl. Algebra, 217 (2013).

C. Năstăsescu, M. van den Bergh, F. vanOystaeyen. Separable functors applied to gradedrings. J. Algebra, 123 (1989)P. Piret. Structure and constructions of cyclicconvolutional codes. IEEE Transactions onInformation Theory, 22 (1976).C. Roos. On the Structure of Convolutional andCyclic Convolutional Codes. IEEE Transactions onInformation Theory, 25 (1979).

J. Gómez-Torrecillas (UGR) CCC & Separable Ring Extensions. EACA2016 (2)

The results are based on

GLN. Generating idempotents inideal codes. ACM Communicationsin Computer Algebra, 48 (2014).

GLN. Convolutional codes with amatrix algebra word-ambient. Adv.Math. Commun, 10(1), 29–43,(2016).

GLN. Separable automorphisms onmatrix algebras over finite fieldsextensions: applications to idealcodes. Proceedings of the 2015ACM on International Symposiumon Symbolic and AlgebraicComputation, 189–195, (2015).


EncodingLet p be a prime number. We work over a finite field F (the alphabet) with q = pa elements.

Information k–tuple Encoded n–tuple Trasmitted n–tuple Decoded n–tuple Recovered k–tupleTransmission DecodingEncoding Post Decoding

Block (linear) EncodingThe encoding process is provided by an injective linear map between finite-dimensional vector spaces, i.e. there isa right invertible matrix G ∈Mk×n(F) such that the encoding is vt = utG ∈ Fn for each information word ut ∈ Fk .ut G vt

The block (linear) code C ⊆ Fn is the vector subspace generated by the rows of G .Cyclic structureAdditional algebraic structure on Fn ∼= F[x ]/〈xn − 1〉 gives cyclic block codes.


EncodingLet p be a prime number. We work over a finite field F (the alphabet) with q = pa elements.Information k–tuple Encoded n–tuple Trasmitted n–tuple Decoded n–tuple Recovered k–tuple

Transmission DecodingEncoding Post Decoding







The block (linear) code C ⊆ Fn is the vector subspace generated by the rows of G .

Cyclic structureAdditional algebraic structure on Fn ∼= F[x ]/〈xn − 1〉 gives cyclic block codes.







Convolutional (linear) CodesConvolutional Encoding

ut

G0

ut−1

G1

⊕

ut−2

· · ·

· · ·

⊕

ut−m

Gm

⊕vt

Delay operatorAlgebraically, the convolutional encoding is given byvt = utG0 + ut−1G1 + · · ·+ ut−mGmIf z represents the delay operator, that is, ut−1 = utz , then we getvt = ut (G0 + zG1 + · · ·+ zmGm)

Thus, the encoder G = G0 + zG1 + · · ·+ zmGm ∈Mk×n(F[z ]).


Convolutional (linear) CodesConvolutional Encoding

ut

G0

ut−1

G1

⊕

ut−2

· · ·

· · ·

⊕

ut−m

Gm

⊕vt

Delay operatorAlgebraically, the convolutional encoding is given byvt = utG0 + ut−1G1 + · · ·+ ut−mGmIf z represents the delay operator, that is, ut−1 = utz , then we getvt = ut (G0 + zG1 + · · ·+ zmGm)

Thus, the encoder G = G0 + zG1 + · · ·+ zmGm ∈Mk×n(F[z ]).J. Gómez-Torrecillas (UGR) CCC & Separable Ring Extensions. EACA2016 (5)

Cyclic structures over Convolutional CodesConvolutional codeThe matrix G is required to be right invertible in Mk×n(F[z ]). A convolutional (linear) code C ⊆ F[z ]n is theF[z ]–submodule generated by the rows of G (which is a direct summand).

Block Code Convolutional CodeC ⊆ Fn , a vector subspace C ⊆ F[z ]n , a direct summand F[z ]–submoduleCyclic Block Code Cyclic Convolutional Code (Naïf)C ⊆ Fn ∼= F [x ]

〈xn−1〉 , an ideal C ⊆ F[z ]n ∼= F [x ]〈xn−1〉 [z ], an ideal that it is a direct summand F[z ]–submodule

Proposition 1 ([Piret’76])An ideal C ⊆ F[z ]n ∼= F [x ]

〈xn−1〉 [z ] that it is a direct summand F[z ]–submodule is generated by vectors in Fn . Thus,Naïf Cyclic Convolutional Codes are Block Codes.

Non Commutative StructuresCyclic structures over convolutional codes non commutative structures on F[z ]n , that is, F[z ]n ∼= F [x ]〈xn−1〉 [z ; σ ][Piret’76, Roos’79, Gluesing and Schmale’04]























Cyclic structures over convolutional codesFor each ring A, the Ore extension A[z ; σ, δ ] is the free right A–module with basis the powers of z andmultiplication defined by the rule az = zσ (a) + δ(a) for all a ∈ R , where σ is a ring endomorphism of A, and δ aσ–derivation.

Let A be a finite (dimensional) algebra of dimension n over the finite field F. Each F–basis B of A becomes anF[z ]-basis of the left F[z ]–module A[z ; σ, δ ], and thus it provides an isomorphism of F[z ]–modulesv : A[z ; σ, δ ]→ F[z ]n . (Note that F[z ] is a subring of A[z ; σ, δ ]).Definition 2 ([Lopez-Permouth and Szabo’13])An ideal code is a left ideal I ≤ A[z ; σ, δ ] such that v(I ) is a direct summand of F[z ]n .We call A is the word–ambient of the convolutional code, while A[z ; σ, δ ] is the sentence–ambient.We focus ourselves in the case δ = 0, i.e. skew polynomials.


Cyclic structures over convolutional codesFor each ring A, the Ore extension A[z ; σ, δ ] is the free right A–module with basis the powers of z andmultiplication defined by the rule az = zσ (a) + δ(a) for all a ∈ R , where σ is a ring endomorphism of A, and δ aσ–derivation.Let A be a finite (dimensional) algebra of dimension n over the finite field F. Each F–basis B of A becomes anF[z ]-basis of the left F[z ]–module A[z ; σ, δ ], and thus it provides an isomorphism of F[z ]–modulesv : A[z ; σ, δ ]→ F[z ]n . (Note that F[z ] is a subring of A[z ; σ, δ ]).

Definition 2 ([Lopez-Permouth and Szabo’13])An ideal code is a left ideal I ≤ A[z ; σ, δ ] such that v(I ) is a direct summand of F[z ]n .We call A is the word–ambient of the convolutional code, while A[z ; σ, δ ] is the sentence–ambient.We focus ourselves in the case δ = 0, i.e. skew polynomials.


Cyclic structures over convolutional codesFor each ring A, the Ore extension A[z ; σ, δ ] is the free right A–module with basis the powers of z andmultiplication defined by the rule az = zσ (a) + δ(a) for all a ∈ R , where σ is a ring endomorphism of A, and δ aσ–derivation.Let A be a finite (dimensional) algebra of dimension n over the finite field F. Each F–basis B of A becomes anF[z ]-basis of the left F[z ]–module A[z ; σ, δ ], and thus it provides an isomorphism of F[z ]–modulesv : A[z ; σ, δ ]→ F[z ]n . (Note that F[z ] is a subring of A[z ; σ, δ ]).Definition 2 ([Lopez-Permouth and Szabo’13])An ideal code is a left ideal I ≤ A[z ; σ, δ ] such that v(I ) is a direct summand of F[z ]n .We call A is the word–ambient of the convolutional code, while A[z ; σ, δ ] is the sentence–ambient.We focus ourselves in the case δ = 0, i.e. skew polynomials.


The generating idempotent problemOne of the advantages of dealing with cyclic block codes is the existence of a generating idempotent (sinceF[x ]/〈xn − 1〉 is semisimple when (p, n) = 1), and that it can be computed.

Questions1 Given an ideal code (i.e., a left ideal) I ⊆ R = A[x ; σ ], is I = Re for some e = e2 ∈ R?2 If yes, can the idempotent e be explicitly computed?

Definition 3An ideal code that it is a direct summand as a left ideal of the sentence ambient R is said to be a split ideal code.Previous: Positive answers to Question 1Every ideal code is split in the following cases:for commutative semisimple ambient word algebras A,[Gluesing and Schmale’04, Lopez-Permouth and Szabo’13]for some automorphisms when the word ambient A is a group algebra of a finite group, in the separable case,[Estrada et al’08, Lopez-Permouth and Szabo’13]


The generating idempotent problemOne of the advantages of dealing with cyclic block codes is the existence of a generating idempotent (sinceF[x ]/〈xn − 1〉 is semisimple when (p, n) = 1), and that it can be computed.Questions

1 Given an ideal code (i.e., a left ideal) I ⊆ R = A[x ; σ ], is I = Re for some e = e2 ∈ R?2 If yes, can the idempotent e be explicitly computed?





Definition 3An ideal code that it is a direct summand as a left ideal of the sentence ambient R is said to be a split ideal code.

Previous: Positive answers to Question 1Every ideal code is split in the following cases:for commutative semisimple ambient word algebras A,[Gluesing and Schmale’04, Lopez-Permouth and Szabo’13]for some automorphisms when the word ambient A is a group algebra of a finite group, in the separable case,[Estrada et al’08, Lopez-Permouth and Szabo’13]






Separable extensionsRecall that the left ideal I is assumed to be an F[z ]–direct summand of R . Thus, an obvious solution to Question 1is to require that the restriction of scalars functor from left R–modules to F[z ]–modules is separable in the senseof cite [Nastasescu/VandenBergh/VanOystaeyen’89].

Definition 4A ring extension S ⊆ R is called separable if the multiplication map µ : R ⊗S R −→ R is a split epimorphism ofR–bimodules, or equivalently if there exists

p =∑i

ei ⊗ fi ∈ R ⊗S R

such thatµ(p) =∑

i

ei fi = 1 and ∀r ∈ R, rp = pr .

This element is called a separability element.Proposition 5 ([Hirata and Sugano’66])In a separable extension, R–submodules which are S–direct summands are also R–direct summands.


Separable extensionsRecall that the left ideal I is assumed to be an F[z ]–direct summand of R . Thus, an obvious solution to Question 1is to require that the restriction of scalars functor from left R–modules to F[z ]–modules is separable in the senseof cite [Nastasescu/VandenBergh/VanOystaeyen’89].Definition 4A ring extension S ⊆ R is called separable if the multiplication map µ : R ⊗S R −→ R is a split epimorphism ofR–bimodules, or equivalently if there exists

p =∑i


such thatµ(p) =∑

i


This element is called a separability element.

Proposition 5 ([Hirata and Sugano’66])In a separable extension, R–submodules which are S–direct summands are also R–direct summands.


Separable extensionsRecall that the left ideal I is assumed to be an F[z ]–direct summand of R . Thus, an obvious solution to Question 1is to require that the restriction of scalars functor from left R–modules to F[z ]–modules is separable in the senseof cite [Nastasescu/VandenBergh/VanOystaeyen’89].Definition 4A ring extension S ⊆ R is called separable if the multiplication map µ : R ⊗S R −→ R is a split epimorphism ofR–bimodules, or equivalently if there exists

p =∑i


such thatµ(p) =∑

i


This element is called a separability element.Proposition 5 ([Hirata and Sugano’66])In a separable extension, R–submodules which are S–direct summands are also R–direct summands.


Separability and skew polynomialsGiven an F–automorphism σ : A→ A, let us denonte σ⊗ = σ ⊗ σ (recall that A denotes a finite F–algebra).Proposition 6The ring extension F[z ] ⊆ A[z ; σ ] is separable if and only if A is separable over F and there exists a separabilityelement p ∈ A⊗F A such that σ⊗(p) = p

Proof.Sufficiency: a straightforward computation shows that any separability element p =∑i ei ⊗F fi ∈ A⊗F A suchthat σ⊗(p) = p gives a separability element ∑i ei ⊗F [z ] fi of the extension F[z ] ⊆ A[z , σ ].Necessity: We have the graded left F[z ]–module A[z ; σ ]⊗F [z ] A[z ; σ ] whose k-th. homogeneous component is(A[z ; σ ]⊗F [z ] A[z ; σ ])

k= F

⟨z ia⊗ z jb | i + j = k, a, b ∈ A

⟩and the isomorphism of graded left F[z ]–modules

φ : A[z ; σ ]⊗F [z ] A[z ; σ ]→ (A⊗F A)[z ; σ⊗]z ia⊗ z jb 7→ z i+j (σ j (a)⊗ b)

This isomorphism allows (‘by taking degree 0 parts’) to get a σ⊗–invariant separability element p of F ⊆ A fromany separability element of F[z ] ⊆ A[z ].



Proof.Sufficiency: a straightforward computation shows that any separability element p =∑i ei ⊗F fi ∈ A⊗F A suchthat σ⊗(p) = p gives a separability element ∑i ei ⊗F [z ] fi of the extension F[z ] ⊆ A[z , σ ].

Necessity: We have the graded left F[z ]–module A[z ; σ ]⊗F [z ] A[z ; σ ] whose k-th. homogeneous component is(A[z ; σ ]⊗F [z ] A[z ; σ ])

k= F

⟨z ia⊗ z jb | i + j = k, a, b ∈ A







k= F

⟨z ia⊗ z jb | i + j = k, a, b ∈ A







k= F

⟨z ia⊗ z jb | i + j = k, a, b ∈ A



This isomorphism allows (‘by taking degree 0 parts’) to get a σ⊗–invariant separability element p of F ⊆ A fromany separability element of F[z ] ⊆ A[z ].J. Gómez-Torrecillas (UGR) CCC & Separable Ring Extensions. EACA2016 (10)

Separable automorphismsSince a finite F–algebra A is separable if and only if A is semi-simple, we may formulate our problem as follows:Separable Automorphism ProblemGiven a semi-simple finite F–algebra A, and an F–automorphism σ : A→ A, is there a separability elementp ∈ A⊗F A such that σ⊗(p) = p? We call such an automorphism separable.

Example 7Let F a field with q elements, and K = Fqt an extension of degree t . Any F–automorphism of K is separable.Indeed, if {αq0 , . . . , αqt−1} and {βq0 , . . . , βqt−1} are dual normal bases, then p =∑i αqi ⊗ βqi ∈ K⊗F K is aninvariant separability element of F ⊆ K for every F–automorphism of K.The Wedderburn-Artin decomposition of A into simple blocks, allows to describe σ in terms of automorphisms ofmatrix rings with coefficients in field extensions of F, so we will concentrate in the matrix case.


Separable automorphismsSince a finite F–algebra A is separable if and only if A is semi-simple, we may formulate our problem as follows:Separable Automorphism ProblemGiven a semi-simple finite F–algebra A, and an F–automorphism σ : A→ A, is there a separability elementp ∈ A⊗F A such that σ⊗(p) = p? We call such an automorphism separable.Example 7Let F a field with q elements, and K = Fqt an extension of degree t . Any F–automorphism of K is separable.Indeed, if {αq0 , . . . , αqt−1} and {βq0 , . . . , βqt−1} are dual normal bases, then p =∑i αqi ⊗ βqi ∈ K⊗F K is aninvariant separability element of F ⊆ K for every F–automorphism of K.

The Wedderburn-Artin decomposition of A into simple blocks, allows to describe σ in terms of automorphisms ofmatrix rings with coefficients in field extensions of F, so we will concentrate in the matrix case.


Separable automorphismsSince a finite F–algebra A is separable if and only if A is semi-simple, we may formulate our problem as follows:Separable Automorphism ProblemGiven a semi-simple finite F–algebra A, and an F–automorphism σ : A→ A, is there a separability elementp ∈ A⊗F A such that σ⊗(p) = p? We call such an automorphism separable.Example 7Let F a field with q elements, and K = Fqt an extension of degree t . Any F–automorphism of K is separable.Indeed, if {αq0 , . . . , αqt−1} and {βq0 , . . . , βqt−1} are dual normal bases, then p =∑i αqi ⊗ βqi ∈ K⊗F K is aninvariant separability element of F ⊆ K for every F–automorphism of K.The Wedderburn-Artin decomposition of A into simple blocks, allows to describe σ in terms of automorphisms ofmatrix rings with coefficients in field extensions of F, so we will concentrate in the matrix case.


Framework and target

F ⊆ K is an extension of finite fields with dual normal bases {αq0 , . . . , αqt−1} and {βq0 , . . . , βqt−1}.Word–ambient is A =Mn(K).The center of an A–bimodule M is MA = {m ∈ M | rm = mr ∀r ∈ A}.

The desired p ∈ A⊗F A must satisfy(P1) p ∈ (A⊗F A)A(P2) p ∈ E1 = {∑i ei ⊗ fi ∈ (A⊗F A)A | ∑i ei fi = 1}

(P3) p ∈ ker(IdA⊗FA − σ⊗)


Framework and target

F ⊆ K is an extension of finite fields with dual normal bases {αq0 , . . . , αqt−1} and {βq0 , . . . , βqt−1}.Word–ambient is A =Mn(K).The center of an A–bimodule M is MA = {m ∈ M | rm = mr ∀r ∈ A}.The desired p ∈ A⊗F A must satisfy(P1) p ∈ (A⊗F A)A(P2) p ∈ E1 = {∑i ei ⊗ fi ∈ (A⊗F A)A | ∑i ei fi = 1

}(P3) p ∈ ker(IdA⊗FA − σ⊗)


(P1) (A⊗F A)AFor each 0 ≤ i , j ≤ n − 1 and all 0 ≤ k ≤ t − 1, we denote

pijk = n−1∑l=0

t−1∑h=0

Eliαqkαqh ⊗ βqhEjl ∈ (A⊗F A)A,where {Eij : 1 ≤ i , j ≤ n} denotes the set of unit matrices of order n.

Lemma 8The canonical projection A⊗F A→ A⊗K A induces an F–linear isomorphism (A⊗F A)A ∼= (A⊗K A)A. As aconsequence, the dimension of (A⊗F A)A as an F–vector space is n2t , and an F–basis for (A⊗F A)A is{pijk | 0 ≤ i , j ≤ n − 1, 0 ≤ k ≤ t − 1}.


(P1) (A⊗F A)AFor each 0 ≤ i , j ≤ n − 1 and all 0 ≤ k ≤ t − 1, we denote

pijk = n−1∑l=0

t−1∑h=0

Eliαqkαqh ⊗ βqhEjl ∈ (A⊗F A)A,where {Eij : 1 ≤ i , j ≤ n} denotes the set of unit matrices of order n.Lemma 8The canonical projection A⊗F A→ A⊗K A induces an F–linear isomorphism (A⊗F A)A ∼= (A⊗K A)A. As aconsequence, the dimension of (A⊗F A)A as an F–vector space is n2t , and an F–basis for (A⊗F A)A is{pijk | 0 ≤ i , j ≤ n − 1, 0 ≤ k ≤ t − 1}.


(P2) E1 = {∑i ei ⊗ fi ∈ (A⊗F A)A | ∑i ei fi = 1}

LetE0 = {p ∈ (A⊗F A)A | µ(p) = 0}andE1 = {p ∈ (A⊗F A)A | µ(p) = 1}.Then E1 is the set of all separability elements of the extension F ⊆ A.

Let p1 = TrK/F (β)∑t−1k=0 p00k ∈ E1.

Proposition 9E0 is an F–vector subspace of (A⊗F A)A and E1 is an affine subspace of (A⊗F A)A both of dimension (n2 − 1)t .An F–basis of E0 is

E = {pijk | 0 ≤ i 6= j ≤ n − 1, 0 ≤ k ≤ t − 1} ∪ {p00k − piik | 1 ≤ i ≤ n − 1, 0 ≤ k ≤ t − 1}.

Moreover E1 = {p1 + q | q ∈ E0}.



LetE0 = {p ∈ (A⊗F A)A | µ(p) = 0}andE1 = {p ∈ (A⊗F A)A | µ(p) = 1}.Then E1 is the set of all separability elements of the extension F ⊆ A. Let p1 = TrK/F (β)∑t−1

k=0 p00k ∈ E1.

Proposition 9E0 is an F–vector subspace of (A⊗F A)A and E1 is an affine subspace of (A⊗F A)A both of dimension (n2 − 1)t .An F–basis of E0 is





LetE0 = {p ∈ (A⊗F A)A | µ(p) = 0}andE1 = {p ∈ (A⊗F A)A | µ(p) = 1}.Then E1 is the set of all separability elements of the extension F ⊆ A. Let p1 = TrK/F (β)∑t−1

k=0 p00k ∈ E1.Proposition 9E0 is an F–vector subspace of (A⊗F A)A and E1 is an affine subspace of (A⊗F A)A both of dimension (n2 − 1)t .An F–basis of E0 is




(P3) ker(IdA⊗FA − σ⊗) IRecall

E0 = {p ∈ (A⊗F A)A | µ(p) = 0}andE1 = {p ∈ (A⊗F A)A | µ(p) = 1}.

A Linear Algebra problem∃p ∈ E1 ∩ ker(IdA⊗FA − σ⊗)KS

��∃q ∈ E0 | σ⊗(p1 + q) = p1 + qKS

��(σ⊗ − IdA⊗FA)(p1) ∈ (IdA⊗FA − σ⊗)(E0)J. Gómez-Torrecillas (UGR) CCC & Separable Ring Extensions. EACA2016 (15)

(P3) ker(IdA⊗FA − σ⊗) IIConverting to matricesWe have F − linear maps K

m

��Mt (F)

f

VV fm = idK

wherem(γ) =

TrK/F (αq0γβq0 ) . . . TrK/F (αq0γβqt−1 )... . . . ...TrK/F (αqt−1γβq0 ) . . . TrK/F (αqt−1γβqt−1 )

f(Γ) = (TrK/F (βq0 ), . . . ,TrK/F (βqt−1 ))Γ(αq0 , . . . , αqt−1

)TThese maps are extended componentwise to (recall that A =Mn(K))

A

m

��Mnt (F)

f

VV fm = idA


Converting to matricesA

σ //

m

��

A

m

��Mnt (F)

f

VV

Mσ//Mnt (F)

f

VV

A⊗F Aσ⊗ //

m⊗

��

A⊗F A

m⊗

��Mnt (F)⊗F Mnt (F) //

f⊗

UU

−�−��

Mnt (F)⊗F Mnt (F)f⊗

UU

−�−��

Mn2t2 (F)Mσ⊗

//Mn2t2 (F)� denotes the Kronecker product of matrices.


Example (Computations done with SAGE) IThe field extension is F = F2 ⊂ F4 = K.The base algebra is A =M2(K).We fix the normal basis B = {a, a2}, which is also self-dual.Two canonical inclusions, A→M4(F) and A⊗F A→M16(F)The basis {pijk} of (A⊗F A)A can now be constructed. For examplep001 = (1 0

0 0

)⊗(a 00 0

)+ (a 00 0

)⊗(a2 00 0

)+ (0 01 0

)⊗(

0 a0 0

)+ (0 0a 0

)⊗(

0 a2

0 0

),

and via the canonical inclusion

p001 =

0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 01 1 0 0 1 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 0 0 1 0 0 0 0 0 0 0 0 0 0 01 0 0 0 0 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 1 0 0 1 1 0 0 0 0 0 0 0 00 0 1 1 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 1 1 0 0 1 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

.


Example (Computations done with SAGE) IIThe basis of E0 is

E = {p010, p011, p100, p101, p000 + p110, p001 + p111}and E1 = p1 + E0 where

p1 = p000 + p001 =

1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 01 0 0 0 0 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 0 0 0 0 1 0 0 0 0 0 0 0 0 0 00 1 0 0 1 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 1 1 0 0 1 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 1 0 0 0 0 0 0 0 00 0 0 1 0 0 1 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

.

The automorphism is σ = σU τ̂ , where U = ( 1 aa2 a

) and τ is the Frobenius automorphism.J. Gómez-Torrecillas (UGR) CCC & Separable Ring Extensions. EACA2016 (19)

Example (Computations done with SAGE) IIIThe matrix Mσ is

Mσ =

0 1 0 1 1 1 1 1 1 0 1 0 0 1 0 11 0 1 0 0 1 0 1 1 1 1 1 1 0 1 01 1 1 0 1 0 0 1 0 1 1 1 1 1 1 00 1 1 1 1 1 1 0 1 0 0 1 0 1 1 10 1 0 1 1 1 1 1 1 0 1 0 0 1 0 11 0 1 0 0 1 0 1 1 1 1 1 1 0 1 01 1 1 0 1 0 0 1 0 1 1 1 1 1 1 00 1 1 1 1 1 1 0 1 0 0 1 0 1 1 11 1 1 1 1 0 1 0 1 1 1 1 1 0 1 00 1 0 1 1 1 1 1 0 1 0 1 1 1 1 11 0 0 1 0 1 1 1 1 0 0 1 0 1 1 11 1 1 0 1 0 0 1 1 1 1 0 1 0 0 11 1 1 1 1 0 1 0 1 1 1 1 1 0 1 00 1 0 1 1 1 1 1 0 1 0 1 1 1 1 11 0 0 1 0 1 1 1 1 0 0 1 0 1 1 11 1 1 0 1 0 0 1 1 1 1 0 1 0 0 1

,

and the matrix Mσ⊗ has size 256× 256.In order to check if (σ⊗− id)(p1) ∈ (id−σ⊗)(E0), we solved the non homogeneous linear system of size 256× 6

v(p1) · (Mσ⊗ − I256) = ∑0≤i 6=j≤1

∑0≤k≤1

αijk(v(pijk ) · (I256 −Mσ⊗ ))+ ∑

0≤i≤1

∑0≤k≤1

αik(v(p00k − piik ) · (I256 −Mσ⊗ ))

whose solution is (0, 0, 1, 0, 1, 0).J. Gómez-Torrecillas (UGR) CCC & Separable Ring Extensions. EACA2016 (20)

Example (Computations done with SAGE) IVThe desired separability element is p = p1 + p100 + p000 + p110 = p100 + p110 + p001. Concretely,

p =

0 1 0 0 1 1 0 0 1 0 0 0 0 1 0 01 1 0 0 1 0 0 0 0 1 0 0 1 1 0 00 0 0 0 0 0 0 0 1 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 1 0 0 1 1 0 01 1 0 0 1 0 0 0 0 1 0 0 1 1 0 01 0 0 0 0 1 0 0 1 1 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 0 1 1 0 00 0 0 0 0 0 0 0 1 1 0 0 1 0 0 00 0 0 1 0 0 1 1 0 0 1 0 0 0 0 10 0 1 1 0 0 1 0 0 0 0 1 0 0 1 10 0 0 0 0 0 0 0 0 0 1 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 1 0 0 1 10 0 1 1 0 0 1 0 0 0 0 1 0 0 1 10 0 1 0 0 0 0 1 0 0 1 1 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1 0 0 1 10 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0

,

Viewed as a tensor product of matrices in M2(K),p = (0 a2

0 0

)⊗(a 00 0

)+ (0 10 0

)⊗(a2 00 0

)+ (0 00 a2

)⊗(

0 a0 0

)+ (0 00 1

)⊗(

0 a2

0 0

)+ (0 a2

0 0

)⊗(

0 0a 0

)+ (0 10 0

)⊗(

0 0a2 0

)+ (0 00 a2

)⊗(

0 00 a

)+ (0 00 1

)⊗(

0 00 a2

)+ (1 0

0 0

)⊗(a 00 0

)+ (a 00 0

)⊗(a2 00 0

)+ (0 01 0

)⊗(

0 a0 0

)+ (0 0a 0

)⊗(

0 a2

0 0

).


Example 10Let A =M2(F2). For this algebra we have thatP = {p00 = ( 1 0

0 0

)⊗(

1 00 0

)+ ( 0 01 0

)⊗(

0 10 0

), p01 = ( 1 0

0 0

)⊗(

0 01 0

)+ ( 0 01 0

)⊗(

0 00 1

),

p10 = ( 0 10 0

)⊗(

1 00 0

)+ ( 0 00 1

)⊗(

0 10 0

), p11 = ( 0 1

0 0

)⊗(

0 01 0

)+ ( 0 00 1

)⊗(

0 00 1

)},

but using the Kronecker product to identify A⊗F2 A with M4(F2), we haveP = {p00 = ( 1 0 0 0

0 0 0 00 1 0 00 0 0 0

), p01 = ( 0 0 0 0

1 0 0 00 0 0 00 1 0 0

), p10 = ( 0 0 1 0

0 0 0 00 0 0 10 0 0 0

), p11 = ( 0 0 0 0

0 0 1 00 0 0 00 0 0 1

)},

and consequentlyE = {p01 = ( 0 0 0 0

1 0 0 00 0 0 00 1 0 0

), p10 = ( 0 0 1 0

0 0 0 00 0 0 10 0 0 0

), p00 − p11 = ( 1 0 0 0

0 0 1 00 1 0 00 0 0 1

)}.

i


Example 11Let us consider the inner automorphism associated to V = ( 1 10 1

). Then

I16 −Mσ⊗V=

0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 0 0 0 0 01 1 0 1 1 1 1 1 0 0 0 0 0 0 0 00 1 0 0 0 1 0 1 0 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 1 1 0 1 0 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 0 0 0 0 01 1 0 0 1 1 0 0 0 1 0 0 1 1 0 00 1 0 0 0 1 0 0 0 0 0 0 0 1 0 01 1 1 1 1 1 1 1 1 1 0 1 1 1 1 10 1 0 1 0 1 0 1 0 1 0 0 0 1 0 10 0 0 0 1 1 0 0 0 0 0 0 0 1 0 00 0 0 0 0 1 0 0 0 0 0 0 0 0 0 00 0 0 0 1 1 1 1 0 0 0 0 1 1 0 10 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0

,

〈G〉 is generated by the rows of (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 0 0 0 1 0 1 0 0 1 0 0 0 1 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

)and

v(p00) · (Mσ⊗V− I16) = (0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0)which does not belong to 〈G〉, and σV is not a separable automorphism.


Separability Test and Cyclic Convolutional Codes.2J. Gómez-Torrecillas ?, F. J. Lobillo ? and G. Navarro ‡

?Department of Algebra and CITIC, University of Granada‡Dep. of Computer Sciences and AI, and CITIC, University of Granada

XV Encuentro de Álgebra Computacional y Aplicaciones, Logroño, June 22th, 2016

1Research supported by grants MTM2013-41992-P and TIN2013-41990-R from Ministerio de Economía y Competitividad of the SpanishGovernment and from FEDER2Research supported by grants MTM2013-41992-P and TIN2013-41990-R from Ministerio de Economía y Competitividad of the SpanishGovernment and from FEDERJ. Gómez-Torrecillas (UGR) CCC & Separable Ring Extensions. EACA2016 (24)

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