Separable Automorphisms on Matrix Algebras over Finite Field Extensions. Applications to Ideal Codes. 1 J. 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 ISSAC 2015, July 9th, 2015 1 Research partially supported by grants MTM2013-41992-P and TIN2013-41990-R from Ministerio de Economía y Competitividad of the Spanish Government and from FEDER F. J. Lobillo (UGR) Separable Automorphisms on Matrix Algebras. ISSAC 2015, July 9th, 2015 (1)
An extremely short overview of coding theory Encoding Decoding Transmission Info. retrieval Information k –tuple Encoded n –tuple Trasmitted n –tuple Decoded n –tuple Recovered k –tuple F. J. Lobillo (UGR) Separable Automorphisms on Matrix Algebras. ISSAC 2015, July 9th, 2015 (2)
An extremely short overview of coding theory Encoding Decoding Transmission Info. retrieval Information k –tuple Encoded n –tuple Trasmitted n –tuple Decoded n –tuple Recovered k –tuple Block (linear) codes The encoding process is provided by a linear map, i.e. there is a right invertible matrix G ∈ M k × n ( F ) such that the encoding is v t = u t G ∈ F n for each information word u t ∈ F k . u t v t G Additional algebraic structure improves encoding and decoding, F n ∼ = F [ x ] /� x n − 1 � , e.g. Reed-Solomon codes. F. J. Lobillo (UGR) Separable Automorphisms on Matrix Algebras. ISSAC 2015, July 9th, 2015 (2)
An extremely short overview of coding theory Encoding Decoding Transmission Info. retrieval Information k –tuple Encoded n –tuple Trasmitted n –tuple Decoded n –tuple Recovered k –tuple Block (linear) codes The encoding process is provided by a linear map, i.e. there is a right invertible matrix G ∈ M k × n ( F ) such that the encoding is v t = u t G ∈ F n for each information word u t ∈ F k . u t v t G Additional algebraic structure improves encoding and decoding, F n ∼ = F [ x ] /� x n − 1 � , e.g. Reed-Solomon codes. Convolutional codes � � � · · · v t G 0 G 1 G m u t u t − 1 u t − 2 · · · u t − m F. J. Lobillo (UGR) Separable Automorphisms on Matrix Algebras. ISSAC 2015, July 9th, 2015 (2)
Convolutional codes and cyclicity Cyclicity in convolutional codes � non commutative structures on F [ z ] n . [Piret’76, Gluesing and Schmale’04, Lopez-Permouth and Szabo’13, GLN’14b] For each ring A , the Ore extension A [ z ; σ, δ ] is the free right A –module with basis the powers of z and multiplication defined by the rule az = z σ ( a ) + δ ( a ) for all a ∈ R , where σ is a ring endomorphism of A , and δ a σ –derivation. Let A be a finite semisimple algebra of dimension n over a finite field F . Each F –basis B of A induces a natural isomorphism of F [ z ]–modules v : A [ z ; σ, δ ] → F [ z ] n . A is the word–ambient of the convolutional code, while A [ z ; σ, δ ] is the sentence–ambient. Definition 1 ([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 focus ourselves in the case δ = 0 , i.e. skew polynomials. F. J. Lobillo (UGR) Separable Automorphisms on Matrix Algebras. ISSAC 2015, July 9th, 2015 (3)
Two questions about ideal codes Are ideal codes direct summands as left ideals? A positive answer is shown in [GLN’14b, GLN ISSAC’14 Poster] if F [ z ] ⊆ A [ z ; σ ] is a separable ring extension. This answer generalizes previous works [Gluesing and Schmale’04] when the word–ambient is a commutative semisimple F –algebra, and [Lopez-Permouth and Szabo’13] whenever it is a separable group algebra of a finite group over a F . F. J. Lobillo (UGR) Separable Automorphisms on Matrix Algebras. ISSAC 2015, July 9th, 2015 (4)
Two questions about ideal codes Are ideal codes direct summands as left ideals? A positive answer is shown in [GLN’14b, GLN ISSAC’14 Poster] if F [ z ] ⊆ A [ z ; σ ] is a separable ring extension. This answer generalizes previous works [Gluesing and Schmale’04] when the word–ambient is a commutative semisimple F –algebra, and [Lopez-Permouth and Szabo’13] whenever it is a separable group algebra of a finite group over a F . Can we compute a generator for an ideal code? In general it is not known if ideal codes are even principal. If the ideal code is a direct summand as left ideal, which we call a split ideal code , then it is principal and generated by an idempotent. This generator can be effectively computed under the separability conditions, see [GLN’14b, Algorithm 1] and [GLN ISSAC’14 Poster, Algorithm 1] F. J. Lobillo (UGR) Separable Automorphisms on Matrix Algebras. ISSAC 2015, July 9th, 2015 (4)
Separable extensions Definition 2 A non commutative ring extension S ⊆ R is called separable if the multiplication map µ : R ⊗ S R − → R � → � i a i ⊗ b i �− i a i b i splits, or equivalently if there exists � p = a i ⊗ b i ∈ R ⊗ S R i such that � µ ( p ) = a i b i = 1 and ∀ r ∈ R , rp = pr . i This element is called a separability element . Proposition 3 ([Hirata and Sugano’66]) In a separable extension, R –submodules which are S –direct summands are also R –direct summands. F. J. Lobillo (UGR) Separable Automorphisms on Matrix Algebras. ISSAC 2015, July 9th, 2015 (5)
Separability and skew polynomials A is a separable F –algebra and σ ∈ Aut F ( A ), σ ⊗ : A ⊗ F A → A ⊗ F A is defined by σ ⊗ ( a ⊗ b ) = σ ( a ) ⊗ σ ( b ). Corollary 4 Let F be a finite field and let A be a separable F –algebra with separability element p = � i a i ⊗ F b i ∈ A ⊗ F A . Let σ ∈ Aut F ( A ) such that σ ⊗ ( p ) = p . Then F [ z ] ⊆ A [ z ; σ ] is a separable extension and a separability element is given by � p = a i ⊗ F [ z ] b i ∈ A [ z ; σ ] ⊗ F [ z ] A [ z ; σ ] . i In particular each ideal code is a split ideal code and it is generated by an idempotent. This corollary follows from [GLN’14b, Theorem 6]. F. J. Lobillo (UGR) Separable Automorphisms on Matrix Algebras. ISSAC 2015, July 9th, 2015 (6)
Computational problems σ ∈ Aut F ( A ) is sepa- rable if ∃ p ∈ A ⊗ F A a separability element such that σ ⊗ ( p ) = p Can be p Is σ a separable effectively automorphism? computed? Each ideal [GLN’14b, code is Algorithm generated by 1] applies an idempotent F. J. Lobillo (UGR) Separable Automorphisms on Matrix Algebras. ISSAC 2015, July 9th, 2015 (7)
Framework and target F ⊆ K is a finite fields extension with dual normal bases { α q 0 , . . . , α q t − 1 } and { β q 0 , . . . , β q t − 1 } . Word–ambient is A = M n ( K ). The center of an A –bimodule M is M A = { m ∈ M | rm = mr ∀ r ∈ A } . The desired p ∈ A ⊗ F A must satisfy (P1) p ∈ ( A ⊗ F A ) A � � � i a i ⊗ b i ∈ ( A ⊗ F A ) A | � (P2) p ∈ E 1 = i a i b 1 = 1 (P3) p ∈ ker ( Id A ⊗ F A − σ ⊗ ) F. J. Lobillo (UGR) Separable Automorphisms on Matrix Algebras. ISSAC 2015, July 9th, 2015 (8)
(P1) ( A ⊗ F A ) A For each 0 ≤ i , j ≤ n − 1 and all 0 ≤ k ≤ t − 1 , we denote n − 1 � � t − 1 E li α q k α q h ⊗ β q h E jl ∈ A ⊗ F A . p ijk = l = 0 h = 0 Lemma 5 The dimension of ( A ⊗ F A ) A as an F –vector space is n 2 t . An F –basis for ( A ⊗ F A ) A is { p ijk | 0 ≤ i , j ≤ n − 1 , 0 ≤ k ≤ t − 1 } . F. J. Lobillo (UGR) Separable Automorphisms on Matrix Algebras. ISSAC 2015, July 9th, 2015 (9)
� � � i a i ⊗ b i ∈ ( A ⊗ F A ) A | � (P2) E 1 = i a i b 1 = 1 Let E 0 = { p ∈ ( A ⊗ F A ) A | µ ( p ) = 0 } and E 1 = { p ∈ ( A ⊗ F A ) A | µ ( p ) = 1 } . Then E 1 is the set of all separability elements of the extension F ⊆ A . Let p 1 = � t − 1 k = 0 Tr K / F ( β ) p 00 k . Proposition 6 E 0 is an F –vector subspace of ( A ⊗ F A ) A and E 1 is an affine subspace of ( A ⊗ F A ) A both of dimension ( n 2 − 1 ) t . An F –basis of E 0 is E = { p ijk | 0 ≤ i � = j ≤ n − 1 , 0 ≤ k ≤ t − 1 } ∪ { p 00 k − p iik | 1 ≤ i ≤ n − 1 , 0 ≤ k ≤ t − 1 } . Moreover E 1 = { p 1 + q | q ∈ E 0 } . F. J. Lobillo (UGR) Separable Automorphisms on Matrix Algebras. ISSAC 2015, July 9th, 2015 (10)
� � � � (P3) ker ( Id A ⊗ F A − σ ⊗ ) I Recall E 0 = { p ∈ ( A ⊗ F A ) A | µ ( p ) = 0 } and E 1 = { p ∈ ( A ⊗ F A ) A | µ ( p ) = 1 } . Linearizing the problem ∃ p ∈ E 1 ∩ ker ( Id A ⊗ F A − σ ⊗ ) ∃ q ∈ E 0 | σ ⊗ ( p 1 + q ) = p 1 + q ( σ ⊗ − Id A ⊗ F A )( p 1 ) ∈ ( Id A ⊗ F A − σ ⊗ )( E 0 ) F. J. Lobillo (UGR) Separable Automorphisms on Matrix Algebras. ISSAC 2015, July 9th, 2015 (11)
� � � � � � � � � � � � � (P3) ker ( Id A ⊗ F A − σ ⊗ ) II Converting to matrices σ A A m f m f � M nt ( F ) M nt ( F ) M σ σ ⊗ A ⊗ F A A ⊗ F A f ⊗ f ⊗ m ⊗ m ⊗ M nt ( F ) ⊗ F M nt ( F ) M nt ( F ) ⊗ F M nt ( F ) − ⊠ − − ⊠ − � M n 2 t 2 ( F ) M n 2 t 2 ( F ) M σ ⊗ F. J. Lobillo (UGR) Separable Automorphisms on Matrix Algebras. ISSAC 2015, July 9th, 2015 (12)
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