Computing Free Distances of Idempotent Convolutional Codes 1 J. Gómez-Torrecillas † , F. J. Lobillo † and G. Navarro ‡ † Department of Algebra and CITIC, University of Granada ‡ Department of Computer Sciences and AI, and CITIC, University of Granada ISSAC 2018, July 17th, 2018 1 Supported by grant MTM2016-78364-P from Agencia Estatal de Investigación (AEI) of the Government of Spain and Fondo Europeo de Desarrollo Regional (FEDER) of the European Union. GLN (UGR) Free distance and ICC ISSAC 2018, July 17th, 2018 (1)
Index Convolutional Codes 1 Free distance 2 Cyclic structures and free distance 3 Computing the free distance 4 GLN (UGR) Free distance and ICC ISSAC 2018, July 17th, 2018 (2)
Index Convolutional Codes 1 Free distance 2 Cyclic structures and free distance 3 Computing the free distance 4 GLN (UGR) Free distance and ICC ISSAC 2018, July 17th, 2018 (3)
Several definitions 픽 a finite field, 픽 [ z ] polynomials in z over 픽 , 픽 ( z ) rational functions, 픽 (( z )) Laurent series. A rate k ∕ n convolutional code can be equivalently defined as ( 픽 ( z ) ) . A k -dimensional vector subspace ≤ 픽 (( z )) n generated by G ( z ) ∈ k × n 1 A k -dimensional vector subspace ≤ 픽 ( z ) n . 2 A rank k direct summand ≤ ⊕ 픽 [ z ] n , i.e. a rank k submodule ≤ 픽 [ z ] n such that 픽 [ z ] n ∕ is torsionfree. 3 Series and polynomials are interesting because they model information and transmitted sequences via the identifications 픽 [ z ] n ≅ 픽 n [ z ] 픽 (( z )) n ≅ 픽 n (( z )) . GLN (UGR) Free distance and ICC ISSAC 2018, July 17th, 2018 (4)
Rational transfer functions Rational functions are interesting because multiplication by f 0 + f 1 z + ⋯ + f m z m 1 + q 1 z + ⋯ + q m z m ∈ 픽 ( z ) corresponds to the rational transfer function ⋯ v t + + + f 0 f 1 f m − 1 f m u t u t − 1 u t − 2 ⋯ u t − m + q 1 q 2 q m ⋯ + + For details, see R. Johannesson and K. Sh. Zigangirov. Fundamentals of Convolutional Coding . Wiley-IEEE Press, 1999 GLN (UGR) Free distance and ICC ISSAC 2018, July 17th, 2018 (5)
Rational functions and polynomials Proposition Let k ≤ n. The map ↦ ∩ 픽 n [ z ] establishes a bijection between the set of k-dimensional vector subspaces of 픽 ( z ) n and the set of all 픽 [ z ] –submodules of 픽 [ z ] n of rank k that are direct summands of 픽 [ z ] n . This proposition is a module-theoretical and coordinate-free refinement of Theorem 3 in G. D. Forney Jr. Convolutional codes I: Algebraic structure. IEEE Transactions on Information Theory 16(6), 720–738, (1970). From now on, a rate k ∕ n convolutional code is a rank k direct summand of 픽 n [ z ] . We also identify ( 픽 [ z ] ) ≅ k × n ( 픽 ) [ z ] . k × n to work with generator (and parity check) matrices. GLN (UGR) Free distance and ICC ISSAC 2018, July 17th, 2018 (6)
Index Convolutional Codes 1 Free distance 2 Cyclic structures and free distance 3 Computing the free distance 4 GLN (UGR) Free distance and ICC ISSAC 2018, July 17th, 2018 (7)
Hamming weight and free distance The Hamming weight w ∶ 픽 n → ℕ is defined as w( v 0 ⋯ v n − 1 ) = | { i | v i ≠ 0 } | . It is a very important parameter for linear block codes (a.k.a. vector subspaces of 픽 n ) because its measures the detection and correction capability of the code. The Hamming weight can be canonically extended to w ∶ 픽 n [ z ] → ℕ ∑ i z i f i ↦ ∑ i w( f i ) . The free distance is defined as d free ( ) = min {w( f ) | f ∈ , f ≠ 0 } = min { w( f ) | f = ∑ i z i f i ∈ , f 0 ≠ 0 } GLN (UGR) Free distance and ICC ISSAC 2018, July 17th, 2018 (8)
Classical row and column distances Introduced by Costello in 1969. For each f = ∑ i z i f i , we denote f [ 0 , j ] = ∑ j i = 0 z i f i . The j th column distance of is defined as j = min { w( f [ 0 , j ] ) | f ∈ , f 0 ≠ 0 } . d c As observed in the proof of [Johannesson and Zigangirov’99, Theorem 3.1], this definition matches with the column distance defined there of any (rational) generator matrix G of such that G ( 0 ) has full rank (e.g. when G is a basic generator matrix). The j th row distance of with respect to a basic generator matrix G = ∑ m i = 0 z i G i , of degree m is defined as d r j = min {w( f ) | f ∈ , f ≠ 0 , deg( f ) ≤ j + m } . This definition depends on the degree m of G as a polynomial in z with matrix coefficients. See [Johannesson and Zigangirov’99, p. 114] for more details. GLN (UGR) Free distance and ICC ISSAC 2018, July 17th, 2018 (9)
Computing the free distance I Theorem ([Johannesson and Zigangirov’99, Ch. 3]) For every index j, d c j ≤ d c j + 1 ≤ d free ( ) ≤ d r j + 1 ≤ d r j , and d c s = d free ( ) = d r s for s big enough. The degree m of G should play some role in row and column distance sequences. In fact, each vector in the information sequence interacts only with the m + 1 coefficients of G . This leads to the following natural question: Does the equality d c j = d c j + m for some j ≥ 0 imply d c j = d free ( ) ? GLN (UGR) Free distance and ICC ISSAC 2018, July 17th, 2018 (10)
Computing the free distance II Example Let be the rate 2 ∕ 4 code generated by the basic matrix ( ) z 4 + z 2 + 1 z 3 + z 2 + z + 1 z 4 + z 3 z 3 + z 2 + z G = . z 4 + z 3 + 1 z 3 z 3 + z + 1 1 With the aid of the computer software SageMath, we have computed the column distances, whose values are written in the following table: j 0 1 2 3 4 5 6 7 8 9 10 11 d c 2 3 4 5 5 6 6 6 6 6 6 7 j So d c 5 = d c 10 = 6, but d free ( ) ≥ 7. Actually, d free ( ) = 8. GLN (UGR) Free distance and ICC ISSAC 2018, July 17th, 2018 (11)
Index Convolutional Codes 1 Free distance 2 Cyclic structures and free distance 3 Computing the free distance 4 GLN (UGR) Free distance and ICC ISSAC 2018, July 17th, 2018 (12)
Naive approach and 휎 -cyclicity Proposition ([Piret’76]) 픽 [ x ] An ideal ⊆ 픽 n [ z ] ≅ ⟨ x n − 1 ⟩ [ z ] that it is a direct summand 픽 [ z ] –submodule is generated by vectors in 픽 n . Thus, Naive Cyclic Convolutional Codes are Block Codes. Cyclic structures over convolutional codes ⇝ non commutative structures on 픽 n [ z ] , that is, 픽 [ x ] 픽 n [ z ] ≅ ⟨ x n − 1 ⟩ [ z ; 휎 ] [Piret’76, Roos’79, Gluesing and Schmale’04] P. Piret. Structure and constructions of cyclic convolutional codes. IEEE Trans. Inform. Theory , 22 (1976). C. Roos. On the Structure of Convolutional and Cyclic Convolutional Codes. IEEE Trans. Inform. Theory , 25 (1979). H. Gluesing-Luerssen and W. Schmale. On cyclic convolutional codes. Acta Appl. Math. , 82 (2004). GLN (UGR) Free distance and ICC ISSAC 2018, July 17th, 2018 (13)
Ideal Codes For each ring A , the Ore extension A [ z ; 휎 ] is the free right A –module with basis the powers of z and multiplication defined by the rule az = z 휎 ( a ) for all a ∈ R , where 휎 is a ring endomorphism of A . Let A be a finite (dimensional) algebra of dimension n over the finite field 픽 . Each 픽 –basis = { b 0 , b 1 , … , b n − 1 } of A becomes an 픽 [ z ] -basis of the left 픽 [ z ] –module A [ z ; 휎 ] , and thus it provides an isomorphism of 픽 [ z ] –modules 픳 ∶ A [ z ; 휎 ] → 픽 n [ z ] . Definition An ideal code is a left ideal I ≤ A [ z ; 휎 ] such that 픳 ( I ) is a direct summand of 픽 n [ z ] . See S. R. López-Permouth and S. Szabo. Convolutional codes with additional algebraic structure. J. Pure Appl. Algebra , 217 (2013). We call A is the word–ambient of the convolutional code, while A [ z ; 휎 ] is the sentence–ambient. GLN (UGR) Free distance and ICC ISSAC 2018, July 17th, 2018 (14)
Idemponent Convolutional Codes I Definition Let R = A [ z ; 휎 ] , and fix a basis of A over 픽 . A convolutional code ⊆ 픽 n [ z ] is said to be an idempotent convolutional code (ICC) if 픳 − 1 ( ) is a direct summand as left ideal of R , i.e. there exists an idempotent 휖 = 휖 2 ∈ R such that 픳 − 1 ( ) = R 휖 . By a slight abuse of language, we will simply say that is generated by 휖 , and we write = R 휖 . The isomorphism 픳 ∶ A → 픽 n associated to allows the extension of the weight from 픽 n to A , i.e. w( a ) = w( 픳 ( a )) for all a ∈ A . This weight is therefore canonically extended to A [ z ; 휎 ] . 휎 is called isometry if w( 휎 ( a )) = w( a ) for all a ∈ A . Examples, as well as algorithms for their construction, of idempotent convolutional codes can be seen in J. Gómez-Torrecillas, F. J. Lobillo, and G. Navarro. Generating idempotents in ideal codes, ACM Communications in Computer Algebra, Vol 48, No. 3, Issue 189, September 2014, ISSAC poster abstracts, pp. 113-115. GLN (UGR) Free distance and ICC ISSAC 2018, July 17th, 2018 (15)
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