A new class of superregular matrices and MDP convolutional codes
aa r X i v : . [ c s . I T ] M a r Abstract
This paper deals with the problem of constructing superregular matrices that lead to MDPconvolutional codes. These matrices are a type of lower block triangular Toeplitz matriceswith the property that all the square submatrices that can possibly be nonsingular due to thelower block triangular structure are nonsingular. We present a new class of matrices that aresuperregular over a sufficiently large finite field F . Such construction works for any given choiceof characteristic of the field F and code parameters ( n, k, δ ) such that ( n − k ) | δ . Finally, wediscuss the size of F needed so that the proposed matrices are superregular. Keywords:
Convolutional codes; Column distances; Maximum distance profile; Superregu-lar matrices.2000 MSC: 94B10, 15B05 new class of superregular matrices and MDPconvolutional codes P. Almeida, D. Napp and R. Pinto
July 24, 2018
In recent years, renewed efforts have been made to further analyze the distance properties of con-volutional codes [2, 4, 5, 6, 9, 13, 14, 15]. Convolutional codes with the maximum possible distance(for a given choice of parameters) are called maximum distance separable (MDS). However, for errorcontrol purposes it is also important to consider codes with large column distances.The convolutional codes whose column distances increase as rapidly as possible for as long aspossible are called maximum distance profile (MDP) codes. These codes are specially appealing forthe performance of sequential decoding algorithms as they have the potential to have a maximumnumber of errors corrected per time interval. In [10] a non-constructive proof of the existence ofsuch codes (for all transmission rates and all degrees) was given. However, the problem of how toconstruct MDP codes is far from being solved and very little is known about the minimum fieldsize required for doing so. It turns out that this issue has been connected to the construction of aparticular type of superregular matrices. In [2] a concrete construction of superregular matrices isgiven for all parameters ( n, k, δ ) although over a field with a large characteristic and size. In [6] thesize of the field needed to have a superregular matrix is studied. They provide a bound on this sizeand conjecture the existence of a much tighter bound based on examples and computer searches.In this paper, we will address these issues and present a new class of matrices that are superregularover a sufficiently large finite field F of any characteristic. We also provide a bound on the requiredfield size needed for such matrices to be superregular. In this section, we recall basic material from the theory of convolutional codes that is relevant tothe presented work and link it to the notion of superregular matrices.In this paper, we consider convolutional codes constituted by codewords having finite support:Let F be a finite field and F [ z ] the ring of polynomials with coefficients in F . A convolutional code C of rate k/n is a F [ z ]-submodule of F [ z ] n of rank k of the form C = im F [ z ] G ( z ) = { G ( z ) u ( z ) : u ( z ) ∈ F k [ z ] } , where G ( z ) ∈ F [ z ] n × k is a right-invertible matrix over F [ z ], i.e., there exists a matrix, called theparity check matrix, H ( z ) ∈ F [ z ] ( n − k ) × n such thatim F [ z ] G ( z ) = ker F [ z ] H ( z ) = { v ( z ) ∈ F [ z ] n : H ( z ) v ( z ) = 0 } . (1)2he degree of C , denoted by δ , is defined as the maximum degree of the full size minors of G ( z ).Notice that we can also choose H ( z ) to be left invertible over F [ z ], and in this case δ will also beequal to the maximum degree of the full size minors of H ( z ). A convolutional code of rate k/n anddegree δ is called an ( n, k, δ ) convolutional code.The most important property of a code is its distance, defined as follows: The weight of apolynomial vector v ( z ) = P i ∈ N v i z i ∈ F [ z ] n is given by wt( v ) = P i ∈ N wt( v i ), where wt( v i ) is thenumber of nonzero elements of v i . The distance of a convolutional code C is defined asd( C ) = min { wt( v ( z )) | v ( z ) ∈ C , v ( z ) = 0 } . If C = ker F [ z ] H ( z ), where H ( z ) = ν X i =0 H i z i , for some ν ∈ N , then the j -th column distance of C isdefined as d cj = min { wt ( v [0 ,j ] ) = wt ( v + v z + · · · + v j z j ) : v ( z ) = X i ∈ N v i z i ∈ C and v = 0 } = min { wt ( ~v j ) : ~v j = [ v . . . v j ] ∈ F ( j +1) n , H ( H , . . . , H j ) ~v ⊤ j = 0 , v ( z ) = X i ∈ N v i z i ∈ C and v = 0 } . where H ( H , . . . , H j ) = H · · · H H · · · H H H · · · H j H j − · · · · · · H ∈ F ( j +1)( n − k ) × ( j +1) n , (2)and H j = 0 for j > ν .In this paper we focus on this important notion of column distance. This notion is closely relatedto the notion of optimum distance profile (ODP), see [7, pp.112]. The following results about columndistances are proved in [2]. Proposition 2.1
Let C be an ( n, k, δ ) convolutional code and L = ⌊ δk ⌋ + ⌊ δn − k ⌋ . Theni) d cj ( C ) ≤ ( j + 1)( n − k ) + 1 , ∀ j ∈ N ;ii) if there exists j ∈ N such that d cj ( C ) = ( j + 1)( n − k ) + 1 , then d ci ( C ) = ( i + 1)( n − k ) + 1 , for i ≤ j and j ≤ L . A convolutional code C is called maximum distance profile ( MDP ) if its column distances achievethe maximum possible values (for a given choice of parameters), i.e., if C has rate k/n and degree δ , then d cL ( C ) = ( L + 1)( n − k ) + 1, for L = ⌊ δ/k ⌋ + ⌊ δ/ ( n − k ) ⌋ and so d cj ( C ) = ( j + 1)( n − k ) + 1,for j ≤ L . In order to characterize MDP codes we need to introduce the notion of superregularmatrices.Let A = [ µ ij ] be a square matrix of order m over F and S m the symmetric group of order m .Recall that the determinant of A is given by | A | = X σ ∈ S m ( − sgn( σ ) µ σ (1) · · · µ mσ ( m ) . (3)A trivial term of the determinant is a term of (3), µ σ (1) · · · µ mσ ( m ) , such that exists 1 ≤ i ≤ m with µ iσ ( i ) = 0. If A is a square submatrix of a matrix B , with entries in F , and all the terms of the3eterminant of A are trivial we say that | A | is a trivial minor of B . We say that B is superregular if all its non-trivial minors are different from zero.It is important to remark here that there exist several related, but different, notions of superreg-ular matrices in the literature. Unfortunately, all these notions are only particular cases of the moregeneral definition given above. Frequently, see for instance [11], a superregular matrix is defined tobe a matrix for which every square submatrix is nonsingular. Obviously all the entries of these ma-trices must be nonzero. Also, in [1, 8, 12], several examples of triangular matrices were constructedin such a way that all submatrices inside this triangular configuration were nonsingular. However,all these notions do not apply to our case as they do not consider submatrices that contain zeros.The more recent contributions [2, 4, 6, 15, 14] consider the same notion of superregularity as us, butdefined only for lower triangular matrices.Next theorem shows how MDP ( n, k, δ ) convolutional codes with ( n − k ) | δ can be characterizedby superregular matrices (see [2, Theorem 3.1]). Theorem 2.1
Let C be an ( n, k, δ ) convolutional code such that ( n − k ) | δ and represented as C =ker F [ z ] [ A ( z ) B ( z )] where A ( z ) = ν X i =0 A i z i ∈ F [ z ] ( n − k ) × ( n − k ) , B ( z ) = ν X i =0 B i z i ∈ F [ z ] ( n − k ) × k , ν = δ ( n − k ) . We can assume without lost of generality that A = I n − k . Furthermore, let A ( z ) − B ( z ) = ∞ X i =0 ¯ H i z i ∈ F (( z )) ( n − k ) × k be the Laurent expansion of A ( z ) − B ( z ) over the field F (( z )) of Laurent series. Define L = ⌊ δ/k ⌋ + δ/ ( n − k ) and b ¯ H = [ I ( L +1)( n − k ) ¯ H ( ¯ H , . . . , ¯ H L )] where ¯ H ( ¯ H , . . . , ¯ H L ) = ¯ H · · · H ¯ H · · · H ¯ H ¯ H · · · ... ... ... ... ... ¯ H L ¯ H L − · · · · · · ¯ H ∈ F ( L +1)( n − k ) × ( L +1) k . (4) The following are equivalent:1. C is MDP.2. ¯ H ( ¯ H , . . . , ¯ H L ) is superregular. Hence, the problem of constructing an MDP convolutional code relies on the problem of constructingsuperregular lower block triangular Toeplitz matrices of the form (4). This problem is addressed inthe next section.For the case where ( n − k ) ∤ δ , similar results were obtained using different methods from systemstheory, see [4, 5, 6] for more details. We will not consider this case in this paper. In this section, we introduce a new class of matrices of the form (4) and show that they are su-perregular matrices over a sufficiently large field F . We conclude the section by providing a lowerbound on the field size of F that ensures the superregularity of the proposed matrices. First, we4ecall previous contributions on superregular matrices.It is a common practice in building the matrix ¯ H ( ¯ H , . . . , ¯ H L ) of Theorem 2.1 to first construct alarge lower triangular superregular matrix in such a way that it contains the lower block triangularToeplitz matrix ¯ H ( ¯ H , . . . , ¯ H L ) as a submatrix. In [2], it was shown that for every positive integer r exists a prime p = p ( r ) such that S r = (cid:18) r − (cid:19) · · · (cid:18) r − (cid:19) (cid:18) r − (cid:19) · · · (cid:18) r − (cid:19) (cid:18) r − (cid:19) (cid:18) r − (cid:19) · · · (cid:18) r − r − (cid:19) (cid:18) r − r − (cid:19) · · · · · · (cid:18) r − (cid:19) (5)is superregular over F p . Moreover, the authors proposed the first rough bound on the size of a field F for a lower triangular Toeplitz matrix A to be superregular over F . Namely if we consider c to be thelargest magnitude among the entries of A and if | F | > c r r r/ , then there exists a superregular lowertriangular Toeplitz matrix A ∈ F r × r . Later, in [6], the following more refined bound was presented:If | F | > B r then there exists a superregular lower triangular Toeplitz matrix A ∈ F r × r , where B r = 12 (cid:18) r (cid:18) r − r − (cid:19) + (cid:18) r − ⌊ r − ⌋ (cid:19)(cid:19) . (6)Moreover, based on examples and computer searches, it was conjectured in [2, 6] that for ℓ ≥ ℓ over the field F ℓ − . If true,it would considerably improve the bound given above. This remains an open problem.We propose a new type of superregular matrices with the form of (4). Of course, this will bringabout a new class of MDP codes. Let ( n, k, δ ) be given such that ( n − k ) | δ . Let M = max { n − k, k } and L = ⌊ δ/k ⌋ + δ/ ( n − k ). Let α be a primitive element of a finite field F = F p N and define[ T | T | . . . | T L ] == α α · · · α M − α M · · · α M − α ML · · · α M ( L +1) − α α · · · α M α M +1 · · · α M α ML +1 · · · α M ( L +1) α α · · · α M +1 α M +2 · · · α M +1 · · · α ML +2 · · · α M ( L +1)+1 ... ... . . . ... ... . . . ... ... . . . ... α M − α M · · · α M − α M − · · · α M − α M ( L +1) − · · · α M ( L +2) − . (7)Define also, T ( T , T , . . . , T L ) ∈ F ( L +1) M × ( L +1) M by T ( T , . . . , T L ) = T · · · T T · · · T T T · · · T L T L − · · · · · · T . (8)We are going to prove that if N is sufficiently large then T ( T , T , . . . , T L ) is superregular. First,we need the following well known result. 5 heorem 3.1 ([3]) Let F be a finite field with p N elements. Let α be a primitive element of F and ρ ( z ) be the minimal polynomial of α (i.e., F = F p [ z ] / ( ρ ( z )) and deg ρ ( z ) = N ). If f ( z ) ∈ F p [ z ] with f ( α ) = 0 then ρ ( z ) | f ( z ) . Theorem 3.2
Let α be a primitive element of a finite field F of characteristic p , ρ ( z ) be the minimalpolynomial of α and consider T ( T , T , . . . , T L ) ∈ F ( L +1) M × ( L +1) M . If | F | ≥ p (2 M ( L +2) − ) then thematrix T ( T , T , . . . , T L ) is superregular (over F ). Proof:
Let [ t L · · · t LM | · · · | t · · · t M | t · · · t M ] denote the columns of T ( T , . . . , T L ) anddefine T ( T , . . . , T L ) = [ t · · · t M | t · · · t M | · · · | t L · · · t LM ], i.e., set T ( T , . . . , T L ) = · · · · · · · · · α · · · α M − · · · · · · · · · α · · · α M ... . . . ... . . . ... . . . ... ... . . . ...0 · · · · · · · · · α M − · · · α M − · · · · · · α · · · α M − α M · · · α M − · · · · · · α · · · α k α M +1 · · · α M ... . . . ... . . . ... . . . ... ... . . . ...0 · · · · · · α M − · · · α M − α M − · · · α M − ... . . . ... . . . ... . . . ... ... . . . ... α · · · α M − · · · α M ( L − · · · α ML − α ML · · · α M ( L +1) − α · · · α M · · · α M ( L − · · · α ML α ML +1 · · · α ML + k ... . . . ... . . . ... . . . ... ... . . . ... α M − · · · α M − · · · α ML − · · · α M ( L +1) − α M ( L +1) − · · · α M ( L +2) − . Next, we show that T ( T , . . . , T L ) is superregular. Obviously, this readily implies that T ( T , . . . , T L )is superregular as well.Let A = [ µ ij ] be a square submatrix of T ( T , . . . , T L ) of order m ≤ M ( L + 1). Note that fromthe particular structure of the proposed matrix T ( T , . . . , T L ) it follows that µ ij ′ ≤ µ ij and µ i ′ j ≤ µ ij if i ′ < i and j ′ < j. (9)Consider m > A as a block matrix in the following form A = O A O A ... O h · · · A h , (10)where, for each 1 ≤ i ≤ h , O i is a null matrix with l i columns and, for each 0 ≤ j ≤ h , A j is amatrix with k j rows and no entry equal to zero. We have l > · · · > l h and m = k > k > · · · > k h . The minor | A | being nontrivial implies k i ≥ l i for any 1 ≤ i ≤ h .Notice that each term of the determinant of A given by (3) is zero or a power of α . We will provethat T ( T , . . . , T L ) is superregular by showing that if there are nontrivial terms in the determinantof A , then there exists a unique term with highest exponent and thus | A | 6 = 0. Let6 = max b ∈ N { b : α b = µ σ (1) · · · µ mσ ( m ) , for some σ ∈ S m } . Thus, it is enough to show that there is a unique σ ∈ S m such that α β = µ mσ ( m ) µ m − σ ( m − · · · µ σ (1) . (11)To this end, we first prove that σ ( m ) is uniquely determined by the following rule: Case 1: If h = 0 or l i < k i for i = 1 , . . . , h , then σ ( m ) = m . Proof Case 1:
Take b σ ∈ S m with µ m b σ ( m ) µ m − b σ ( m − · · · µ b σ (1) = 0 , with µ m b σ ( m ) = µ mm . Let b β ∈ N , such that α b β = µ m b σ ( m ) µ m − b σ ( m − · · · µ b σ (1) . Since h = 0 or l j < k j for all j = 1 , . . . , h then µ ( m − i ) i = 0 for any i = 1 , . . . , m − i , µ ( m − i ) i = 0 then there exists j ∈ { , . . . , h } such that l j ≥ i and k j ≤ i , a contradiction).Construct e σ ∈ S m recursively, as follows:1. Define δ = m and while µ b σ − ( δ i ) b σ ( m ) = 0, let δ i +1 = b σ (cid:18) max j ≥ m − b σ − ( δ i ) b σ − ( j ) (cid:19) . Let i be the first integer such that µ b σ − ( δ i ) b σ ( m ) = 0;2. e σ ( m ) = m and e σ ( b σ − ( δ m )) = b σ ( m );3. For 1 ≤ i ≤ i , e σ ( b σ − ( δ i )) = δ i +1 ;4. For i I = { b σ − ( δ i ) | i = 1 , . . . , i } , e σ ( i ) = b σ ( i ).Clearly, b σ − ( δ i ) > b σ − ( δ i − ) and by (9), µ m b σ ( m ) i Y i =1 µ b σ − ( δ i ) δ i ≤ µ mm , Therefore α b β = Y i ∈ I µ i b σ ( i ) Y i I µ i b σ ( i ) ≤ µ mm Y i I ∪{ m } µ i b σ ( i ) < µ m e σ ( m ) Y i I ∪{ m } µ i e σ ( i ) Y i ∈ I µ i e σ ( i ) which implies that b β is not a maximum, that is b β < β .7 ase 2: If l i = k i for some i ∈ { , . . . , h } , then σ ( m ) = l i , where i is the minimum i ∈ { , . . . , h } such that l i = k i . Proof Case 2:
Take b σ ∈ S m with µ m b σ ( m ) µ m − b σ ( m − · · · µ b σ (1) = 0 , and µ m b σ ( m ) = µ ml i . Let α b β = µ m b σ ( m ) µ m − b σ ( m − · · · µ b σ (1) . It is clear that in this case b σ ( m ) b σ ( m − · · · b σ ( m − l i + 1) necessarily belong to the set { , . . . , l i } .Hence, one can consider the matrix A ′ form by the first l i columns and the last l i rows of A . Applyingthe previous reasoning it is straightforward to see that we are now in the situation of the case 1 forthe new matrix A ′ . Thus, µ mσ ( m ) = µ ml i .Once σ ( m ) has been uniquely determined, we can remove from A its m -th row and its σ ( m )-thcolumn to obtain a new square matrix A of order m −
1. We follow the same previous argumentsapplied to A instead of to A to determine σ ( m − σ ∈ S m and therefore prove the existence of a unique maximum in the terms of (3).Next, we prove the bound on the size of a field F in order to T ( T , . . . , T L ) to be superregularover F .We just proved that there exists a unique term with highest exponent in each nontrivial deter-minant of every submatrix A = [ µ ij ] of T ( T , T , . . . , T L ). Let α β = µ ( r ) σ ( r ) µ ( r − σ ( r − · · · µ σ (1) (12)for some σ ∈ S r with 2 ≤ r ≤ ( L + 1) M , be the highest term one can find. Define R k = { ( i, j ) ∈ N | ≤ i, j ≤ k and i = k or j = k } and R ( σ ) = { ( i, j ) ∈ N | ( i, j ) = ( t, σ ( t )) for some t ∈{ , , . . . , r }} . It follows from the properties of the matrix T ( T , T , . . . , T L ) that Y ( i,j ) ∈ R k ∩ R ( σ ) µ ij ≤ α M ( L +2) − r − k +1) for k = 1 , . . . , r . Hence, α β = r Y k =1 Y ( i,j ) ∈ R k ∩ R ( σ ) µ ij ≤ α M ( L +2) − α M ( L +2) − · · · α M ( L +2) − r < α (2 M ( L +2) − )( P ∞ i =0 − i ) = α (2 M ( L +2) − )( ) < α M ( L +2) − . So β < m ( L +2) − . This means that the maximum exponent of α appearing in the determinants ofthe submatrices of T ( T , T , . . . , T L ) is upper bounded by 2 M ( L +2) − .Notice that deg( ρ ( z )) ≥ M ( L +2) − . If T ( T , T , . . . , T L ) is not superregular over F then thereexists a nontrivial determinant f ( α ) = M ( L +2) − X i =0 ǫ i α i , ǫ i ∈ { , , , . . . , p − } of a submatrix A of T ( T , T , . . . , T L ) such that f ( α ) = 0. By Theorem 3.1 it follows that ρ ( z ) | f ( z ) which contradictsthe fact that the degree of f ( z ) is less than 2 M ( L +2) − . (cid:3) Remark 3.1
Note that if A is a submatrix of a matrix B of the form (4) or (8) then | A | is a trivialminor of B if it contains zeros in its diagonal, and therefore in order to check the superregularity of it is enough to verify the determinant of the submatrices with no zero elements in its diagonal.More concretely, it can be checked that if A = [ µ ij ] is a square submatrix of order r of a matrix ofthe form (4) or (8) then | A | is a nontrivial determinant if and only if its indices i < i < · · · p ((2 M ( L +2) − )( )) in order to T ( T , T , . . . , T L ) to be superregular. It can be checked using computer algebra programs that thereare particular examples (for small values of ( n, k, δ )) of superregular matrices that require a muchsmaller field size, see for instance [2, Example 3.10]. However, the proposed superregular matricescan be constructed for any given characteristic p and parameters ( n, k, δ ) and therefore provides ageneral construction. Note that the superregular matrix S r given in (5) requires, in general, a largecharacteristic p ( r ).We are now in the position to present a new class of MDP convolutional codes. The result easilyfollows from Theorem 2.1, Theorem 3.2 and the fact that submatrices of a superregular matrixinherit the superregularity property. Corollary 3.1
Let ( n, k, δ ) be given and let T ℓ = [ t ℓij ] , i, j = 1 , , . . . , m and ℓ = 0 , , . . . , L bethe entries of the matrix T ℓ as in (7). Define ¯ H ℓ = [ t ℓij ] i = 1 , , . . . , ( n − k ) , j = 1 , , . . . , k and ℓ = 0 , , . . . , L . If | F | ≥ p (2 M ( L +1)+ n − ) then, the convolutional code C = ker F [ z ] [ A ( z ) B ( z )] where A ( z ) = ν X i =0 A i z i ∈ F [ z ] ( n − k ) × ( n − k ) and B ( z ) = ν X i =0 B i z i ∈ F [ z ] ( n − k ) × k , with ν = δn − k , A = I n − k , A i ∈ F ( n − k ) × ( n − k ) , i = 1 , . . . , ν obtained by solving the equations [ A ν · · · A ] ¯ H L − ν · · · ¯ H ¯ H L − ν +1 · · · ¯ H ... ... ¯ H L − · · · ¯ H ν = − [ ¯ H L · · · ¯ H ν +1 ] , and B i = A ¯ H i + A ¯ H i − + · · · + A i ¯ H , i = 0 , . . . , ν, is an MDP convolutional code of rate k/n anddegree δ . Remark 3.2
Details about the construction of the matrices A ( z ) and B ( z ) presented in Corollary3.1 can be found in [2, Appendix C] The following example illustrates the construction of a (5 , ,
3) MDP convolutional code.
Example 3.1
Since n = 5 , k = 2 and δ = 3 , we have that L = 2 and ν = 1 . Let us consider α aroot of the primitive polynomial x + x + x + x + 1 ∈ F [ x ] , i.e., a primitive element overthe field F and the matrix [ ¯ H ¯ H ¯ H ] = α α | α α | α α α α | α α | α α α α | α α | α α over F . Considering A ( z ) = I + A z such that A ¯ H = − ¯ H , where, a possible choice is A ( z ) = + α − α − α +2 + α +2 − α +2 + α +2 − α +2 + α +2 α +2 − α +2 α +2 − α +2 α +2 − α +2 − α +2 z, nd B ( z ) = B + B z such that B = ¯ H and B = ¯ H + A ¯ H , we have that C = ker F [ z ] [ A ( z ) B ( z )] is a (5 , , MDP convolutional code.
There is a type of superregular matrices that are essential for the construction of MDP convolutionalcodes. However, very little is understood about how to construct these matrices and how large afinite field must be, so that a superregular matrix of a given order can exist over that field. In thispaper, we have presented a new class of MDP ( n, k, δ ) convolutional codes, such that ( n − k ) | δ , bymeans of the construction of a novel type of superregular matrices over a field of any characteristic.We also established a bound for the size of the field needed for these matrices to be superregular. References [1] A. K. Aidinyan.
On Matrices with Nondegenerate Square Submatrices . Probl. Peredachi Inf., (4): 106-108, 1986.[2] H. Gluesing-Luerssen, J. Rosenthal and R. Smarandache. Strongly MDS convolutional codes .IEEE Trans. Inf. Th., (2): 584-598, 2006.[3] T.W. Hungerford. Algebra . Springer-Verlag, New York, 1974.[4] R. Hutchinson
The Existence of Strongly MDS Convolutional Codes . SIAM Journal on Controland Optimization, (6): 2812-2826, 2008.[5] R. Hutchinson, J. Rosenthal and R. Smarandache. Convolutional codes with maximum distanceprofile . Systems & Control Letters, (1): 53-63, 2005.[6] R. Hutchinson, R. Smarandache and J. Trumpf. On superregular matrices and MDP convolu-tional codes . Linear Algebra and its Applications, : 2585-2596, 2008.[7] R. Johannesson and K.S. Zigangirov.
Fundamentals of Convolutional Coding . IEEE Press Seriesin Digital and Mobile Comm., 1999.[8] F. J. MacWilliams and N. J. A. Sloane.
The theory of error-correcting codes. II . North-HollandPublishing Co., Amsterdam, 1977. North-Holland Mathematical Library, Vol. 16.[9] J. M. Mu˜noz Porras, J. A. Dom´ınguez P´erez, J. I. Iglesias Curto and G. Serrano Sotelo.
Convolutional Goppa codes . IEEE Trans. Inf. Th., (1), 340-344, 2006.[10] J. Rosenthal and R. Smarandache. Maximum distance separable convolutional codes . Appl.Algebra Engrg. Comm. Comput., (1), 15-32, 1999.[11] R. M. Roth and A. Lempel. On MDS codes via Cauchy matrices . IEEE Trans. Inf. Th., (6), 1314-1319, 1989.[12] R. M. Roth and G. Seroussi. On generator matrices of MDS codes . IEEE Trans. Inf. Th., (6), 826-830, 1985.[13] R. Smarandache, H. Gluesing-Luerssen and J. Rosenthal. Constructions of MDS-convolutionalcodes . IEEE Trans. Inf. Th., (5), 2045-2049, 2001.1014] V. Tom´as. Complete-MDP Convolutional Codes over the Erasure Channel . Departamento deCiencia de la Computaci´on e Inteligencia Artificial, Universidad de Alicante, Alicante, Espa˜na.Jul. 2010.[15] V. Tom´as, J. Rosenthal and R. Smarandache.