Non-catastrophicity in multidimensional convolutional coding

Discrete Mathematics 343 (2020) 111789

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Non-catastrophicity in multidimensional convolutional coding Vakhtang Lomadze Department of Mathematics, I. Javakhishvili Tbilisi State University, Tbilisi 0183, Georgia

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Article history: Received 9 March 2019 Received in revised form 30 October 2019 Accepted 16 December 2019 Available online 14 January 2020

a b s t r a c t The property of non-catastrophicity of multidimensional convolutional codes is studied. In particular, an algebraic and system-theoretic characterization of non-catastrophicity is offered in the multidimensional setting, and the Massey–Sain classical criterion is extended to this setting. © 2019 Elsevier B.V. All rights reserved.

Keywords: Modules Convolutional codes Non-catastrophicity Polynomials Formal series Linear systems

1. Introduction Multidimensional convolutional codes generalize naturally classical one-dimensional ones. However, in contrast to the one-dimensional case, there has been a little of research work in the field of multidimensional convolutional codes. Papers are devoted to this field (of which we are aware) are: Charoenlarpnopparut [2], Climent et al. [3], Fornasini and Valcher [4], Gluesing-Luerssen et al. [5], Jangisarakul and Charoenlarpnopparut [7], Kitchens [8], Lomadze [12], Napp Avelli et al. [14,15], Valcher and Fornasini [18], Weiner [19], Zerz [20]. The present article concerns with the property of non-catastrophicity for arbitrary (not necessarily free) multidimensional convolutional codes. The concept was originally introduced by Massey for usual convolutional codes and it is of great importance. In a catastrophic code it is possible for a ‘‘finite’’ error in the code sequence to result an ‘‘infinite’’ error in the corresponding input sequence. For this reason, catastrophicity is an undesirable property, and convolutional codes with this property should be avoided. The article is organized as follows: Section 1 is a preliminary section, where the Lefschetz duality between polynomials and formal series is discussed. In Section 2, we carefully define non-catastrophicity in the general n-D case. In Section 3, we derive a necessary and sufficient condition for a code to be non-catastrophic in terms of torsion-freeness. In Section 4, we briefly recall the duality between convolutional codes and linear systems, and then find out what does mean noncatastrophicity in terms of linear systems. In the last section, we present a non-catastrophicity criterion that can be regarded as a generalization of the classical Massey–Sain criterion. Throughout, F is a (finite) field, n is a positive integer and s1 , . . . , sn , t1 , . . . , tn are indeterminates satisfying the relations s1 t1 = 1, . . . , sn tn = 1. We put s = (s1 , . . . , sn ) and t = (t1 , . . . , tn ). A multi-index is an n-tuple of nonnegative E-mail address: [email protected]. https://doi.org/10.1016/j.disc.2019.111789 0012-365X/© 2019 Elsevier B.V. All rights reserved.

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V. Lomadze / Discrete Mathematics 343 (2020) 111789

integers, i.e., an element of Zn+ . There is an evident componentwise ordering on the set of multi-indices. For a multi-index i = (i1 , . . . , in ), define |i| = i1 + · · · + in and write i

i

si = s11 . . . sinn , t i = t11 . . . tnin . We shall deal with the following four rings:

F[s], F[[t ]], F[[s]] and F[t ]. (One bracket is used to denote the ring of polynomials and two brackets for the ring of formal series.) Among these rings the first two are especially important. Elements of F[s] are called words and elements of F[[t ]] trajectories. (Elements of F[[s]] can be interpreted as words that may have infinite length and elements of F[t ] as finite trajectories.) On F[[t ]] we consider partial backward-shift operators σ1 , . . . , σn . For each k = 1, . . . , n, the operator σk is defined by the formula

σk (

∑

bi t i ) =

i≥0

∑

bi+ek t i ,

i≥0

where ek denotes the multi-index with 1 in the kth position and all other entries zero. These operators allow to regard F[[t ]] as a module over F[s]. Notice that F[t ] is shift-invariant, and therefore also can be viewed as a module over F[s]. It is useful to have in disposal the ring F((t)), the ring of Laurent formal series in t. It consists of those elements in the quotient field of F[[t ]] that have the form si g with i ∈ Zn+ and g ∈ F[[t ]]. Obviously,

F[s] ⊆ F((t)) and F[[t ]] ⊆ F((t)). Putting σ = (σ1 , . . . , σn ) and letting Π denote the canonical projection map of F((t)) onto F[[t ]], we have: f (σ )g = Π (fg) ∀f ∈ F[s], g ∈ F[[t ]]. For each k ≥ 0, we let F[s]≤k and F[t ]≤k denote the spaces of polynomials in s and t, respectively, of degree ≤ k. Fix once for all a positive integer q. Following Weiner [19], we define a convolutional code (of length q) as a submodule of F[s]q . This definition generalizes the one given in Rosenthal et al. [16] for 1-dimensional case. It is interesting to note that Fornasini and Valcher [4,18] introduce a convolutional code as a submodule of F[s, t ]q . If A ⊆ F[s]q is a convolutional code, there exist some positive integer p and a matrix G such that A = GF[s]p . We call such a matrix a generator matrix of A. 2. Preliminaries on duality Here, we discuss the needed special case of the general duality for vector spaces developed by Lefschetz in [10, Ch.II]. (Lefschetz’ duality generalizes the classical duality of finite-dimensional vector spaces.) Recall that the dual of a topological vector space E, which we denote by E ′ , is the space of all continuous linear forms on E. There are canonical bilinear forms

F[s] × F[[t ]] → F, F[s] × F[t ] → F, F[[s]] × F[t ] → F. They all are given by the formula

⟨

∑

ai s i ,

i

∑

bj t j ⟩ =

j

∑

ai b j .

i=j

All these bilinear forms are non-degenerate. The bilinear forms above induce, in an obvious way, the bilinear forms

F[s]q × F[[t ]]q → F, F[s]q × F[t ]q → F, F[[s]]q × F[t ]q → F, respectively. These are given by the formula

⟨

∑ i

ai s i ,

∑ j

bj t j ⟩ =

∑

atr i bj ,

i=j

and also are non-degenerate, of course. (Here and below ‘‘tr’’ stands for the transpose.) Remark. The above bilinear forms can be introduced also by the formula

⟨f , g ⟩ = (f tr (σ )(g))(0). There is a standard topology on F[s], which is defined as follows: fk → f means that there exists N such that all fk and f belong to F[s]≤N and the coefficient of si in fk converges to the coefficient of si in f for every i with 0 ≤ |i| ≤ N. We consider on F[[t ]] the topology of simple convergence, in which gk → g means that the coefficient of t i in gk converges to the coefficient of t i in g for every i. (On F[[s]] and F[t ], one has similar topologies, of course.)

V. Lomadze / Discrete Mathematics 343 (2020) 111789

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Lemma 1. The bilinear form F[s] × F[[t ]] → F puts F[s] and F[[t ]] in duality, i.e., gives rise to isomorphisms

F[s] ≃ F[[t ]]′

and

F[[t ]] ≃ F[s]′ .

Proof. A fundamental system of neighborhoods of 0 in F[[t ]] is given by the sets

∑

Uk = {

bj t j ∈ F[[t ]] : bj = 0 ∀j such that |j| ≤ k},

j

where k runs over all nonnegative integer numbers. A linear form on F[[t ]] is continuous if and only if it vanishes on Uk for some k ≥ 0. Continuous linear forms that are zero on Uk can be identified with linear forms on

F[t ]≤k = F[[t ]]/Uk . Linear forms on F[t ]≤k , in turn, can be identified with polynomials in s of degree ≤ k, i.e., with elements of F[s]≤k . As F[s] is a union of all F[s]≤k , the first isomorphism follows. The proof of the second one goes as follows. A map from F[s] is continuous if and only if its restriction on each F[s]≤k is continuous. All linear forms on a finite-dimensional space are continuous, and therefore all linear forms on F[s] automatically are continuous. Now, assume we have a linear form L on F[s]. It can be identified with a compatible family of linear forms Lk : F[s]≤k → F. The canonical bilinear form F[s]≤k × F[t ]≤k → F allows to identify Lk with a certain polynomial gk ∈ F[t ]≤k , and thus L gives rise to the infinite sequence (g0 , g1 , g2 , g3 , . . .) of polynomials. These polynomials are compatible in the sense that, for each k ≥ 1, the canonical map F[t ]≤k → F[t ]≤k−1 sends gk to gk−1 . It remains to see that there is a unique formal series g ∈ F[[t ]] whose truncations are the polynomials gk . The correspondence L ↦ → g clearly is bijective. The proof is complete. □ Remark. A similar statement holds, of course, for the bilinear form F[[s]] × F[t ] → F. It induces the isomorphisms

F[t ] ≃ F[[s]]′

and

F[[s]] ≃ F[t ]′ .

The sets Uk defined above are convex, and therefore F[[t ]] is a locally convex space. (So is F[[s]], of course.) For locally convex (topological vector) spaces, one has the Hahn–Banach theorem, which is a central fact in functional analysis. It states (see Theorem 3.5 in Rudin [17]) that if E is a locally convex space, and if X is a subspace of E and y a point of E not in the closure of X , then there exists a continuous linear functional L on E such that L|X = 0

and L(y) ̸ = 0.

3. Definition of non-catastrophicity Given a subset X ⊆ F[s]q , let X denote the closure of X in F[[s]]q . This is obtained from X by means of limit operation, namely, w ∈ F[[s]]q belongs to X if and only if there is a sequence w1 , w2 , w3 , . . . in X converging to w in the sense of the topology of simple convergence. Notice that if A ⊆ F[s]q is a convolutional code, then A is the smallest F[[s]]-submodule in F[[s]]q containing A. Warning. In this section (and the next one as well), the symbol ‘‘⊥’’ will be used exclusively to denote the orthogonal complement with respect to the bilinear form

F[s]q × F[t ]q → F.

(1)

The following two inclusions A ⊆ F[s]q ∩ A and A ⊆ A⊥⊥ are obvious. Lemma 2. Let X ⊆ F[s]q be an F-linear subspace. Then, the double orthogonal complement of X with respect to the bilinear form

F[[s]]q × F[t ]q → F

(2)

is equal to X . Proof. The orthogonal complement of X with respect to (2) is the same as that with respect to (1), i.e., X ⊥ . Let X1 denote the orthogonal complement of X ⊥ with respect to (2). Clearly, X ⊆ X1 . Any orthogonal complement is closed, and consequently X1 is closed. It follows that X ⊆ X1 . Assume now that there is x ∈ X1 that does not belong to X . Since F[[s]] is locally convex, the Hahn–Banach theorem applies, and we find that there exists a continuous linear form y on F[[s]]q such that y|X = 0

and y(x) ̸ = 0.

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V. Lomadze / Discrete Mathematics 343 (2020) 111789

By the remark made in the previous section, F[[s]]′ ≃ F[t ], and so we can view y as an element of F[t ]q and rewrite the conditions above as and ⟨x, y⟩ ̸ = 0.

y ∈ X⊥

We see that x ∈ / X1 . A contradiction. The lemma is proved. □ Corollary 1. Let A ⊆ F[s]q be a convolutional code. Then, A⊥⊥ = A ∩ F[s]q . Proof. It is clear that if Y is a subset in F[t ]q and if X is its orthogonal complement with respect to (2), then Y ⊥ = X ∩ F[s]q . This implies what we want. □ Corollary 2. Let A ⊆ F[s]q be a convolutional code. The following two conditions are equivalent: (a) A = F[s]q ∩ A; (b) A = A⊥⊥ . Definition. A convolutional code A is non-catastrophic if it satisfies the equivalent conditions of Corollary 2. Thus, A is non-catastrophic if the set of finite length codewords of A is precisely A. Equivalently, A is non-catastrophic if the double orthogonal complement of A coincides with A. It is worth noting that non-catastrophicity is an internal property of a convolutional code. (This is by the very definition.) A polynomial matrix G of size q × p is said to be non-catastrophic if GF[s]p = F[s]q ∩ GF[[s]]p . Proposition 1. Let A ⊆ F[s]q be a convolutional code, and let G be any its generator matrix. Then, A is non-catastrophic if and only if so is G. Proof. This is obvious; this follows from the fact that A = GF[[s]]p . (Here p denotes the column number of G.)

□

4. Non-catastrophicity as a kind of torsion-freeness To begin with, recall that a module Q over a (unit ring) is called injective if, for any submodule N ⊆ M and any homomorphism ϕ : N → Q , there exists a homomorphism ψ : M → Q such that ψ|N = ϕ . The following result is a simple consequence of the local duality theory (see Grothendieck [6, Ch. III]). For convenience of readers, we give a direct proof. Lemma 3. F[t ], as a module over F[s], is injective. Proof. Let m denote the ideal of F[s] generated by the indeterminates s1 , . . . , sn . Note that, for each nonnegative integer k, the power mk is the ideal generated by the elements si with |i| = k. First, we need to show that if M is an F[s]-module with the property that mk M = 0 for some k, then there is a canonical isomorphism HomF[s] (M , F[t ]) ≃ HomF (M , F). Given φ ∈ HomF[s] (M , F[t ]) define φ˜ to be the linear map a ↦ → the constant term of φ (a). For u ∈ HomF (M , F), let u¯ be the homomorphism a ↦→

∑

u(si a)t i .

i≥0

(The sum here is finite since si a = 0 for all multi-indices i for which |i| ≥ k.) One easily checks that the maps

φ ↦→ φ˜

and

u ↦ → u¯

are inverse to each other.

V. Lomadze / Discrete Mathematics 343 (2020) 111789

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To prove the injectivity, use Baer’s criterion. Let I ⊆ F[s] be an ideal and let u : I → F[t ] be an F[s]-homomorphism. Since I is finitely generated, we can find an integer k ≥ 0 such that mk annihilates Im(u). Hence, u vanishes on the ideal mk I. Further, by the Artin–Rees lemma (see Bourbaki [1, Ch. III, § 3]), there exists an integer l ≥ 0 such that

ml ∩ I ⊆ mk I . It follows that u vanishes on ml ∩ I, and therefore induces a canonical homomorphism from I /(ml ∩ I) to F[t ]. We have a canonical embedding I /(ml ∩ I) → F[s]/ml . Applying the exact functor HomF (−, F) to it, we get a surjective linear map HomF (F[s]/ml , F) → HomF (I /(ml ∩ I), F). In view of the above canonical isomorphism, this yields a surjective homomorphism HomF[s] (F[s]/ml , F[t ]) → HomF[s] (I /(ml ∩ I), F[t ]). Thus, we can extend the homomorphism I /(ml ∩ I) → F[t ] to a homomorphism F[s]/ml → F[t ]. The composition

F[s] → F[s]/ml → F[t ] is a homomorphism that extends the given homomorphism u. The proof is complete. □ Lemma 4. Let M be a module such that polynomials with nonzero constant term are not zero divisors on M. Then, for every nonzero element x ∈ M, there exists a homomorphism φ : M → F[t ] such that φ (x) ̸ = 0. Proof. Let I = F[s]x, and define a homomorphism φ : I → F[t ] by the formula φ (fx) = f (0). This is well-defined. Indeed, if fx = 0, then f (0) = 0 necessarily, and consequently φ (fx) = 0. Next, we have: φ (x) = 1. Because F[t ] is an injective module, this can be extended to a homomorphism M → F[t ]. The proof is complete. □ Theorem 1. A convolutional code A ⊆ F[s]q is non-catastrophic if and only if polynomials λ ∈ F[s] with λ(0) ̸ = 0 are not zero divisors on F[s]q /A. Proof. ‘‘If’’ We have to show that A⊥⊥ ⊆ A. Assume that h is an element in F[s]q that does not belong to A. Then, by the previous lemma, there exists a homomorphism

F[s]q /A → F[t ] taking the coset of h to a nonzero element. In other words, there is a homomorphism ϕ : F[s]q → F[t ] that is zero everywhere on A, but not on h. At least one term in ϕ (h) is distinct from 0. Multiplying ϕ , if necessary, by some si , we may assume that its constant term (ϕ (h))(0) ̸ = 0. Now, any homomorphism F[s]q → F[t ] is of the form f ↦ → f tr (σ )ξ with ξ ∈ F[t ]q . So, there is ξ ∈ F[t ]q such that

ϕ (f ) = f tr (σ )ξ ∀f ∈ F[s]q . We then have: f tr (σ )ξ = 0 ∀f ∈ A and ⟨h, ξ ⟩ ̸ = 0. We see that ξ ∈ A⊥ and that h ∈ / A⊥⊥ . ‘‘Only if’’ This part is easy. Indeed, by the hypothesis, the canonical map

F[s]q /A → F[[s]]q /A is injective. This allows to view F[s]q /A as a submodule of F[[s]]q /A. Now, if λ is a polynomial with nonzero constant term, then it is invertible as an element of F[[s]]. Hence, it cannot be a zero divisor on F[[s]]q /A. It cannot be a zero divisor on every submodule of this latter, as well. The theorem is proved. □ The theorem can be reformulated as follows: A convolutional code A is catastrophic if and only if there exist a polynomial λ ∈ F[s] with λ(0) ̸ = 0 and a word a ∈ F[s]q such that

λa ∈ A and a ∈ / A. Another equivalent formulation of the above theorem is: A convolutional code A ⊆ F[s]q is non-catastrophic if and only if all irreducible polynomials λ ∈ F[s] with λ(0) ̸ = 0 are not zero divisors on F[s]q /A.

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V. Lomadze / Discrete Mathematics 343 (2020) 111789

Example 1. Let n = 2, and let A = s1 F[s] + s2 F[s]. Then A = {x ∈ F[s] | x(0) = 0}, and it is clear that if λ ∈ F[s] is such that λ(0) ̸ = 0, then

λa ∈ A ⇒ a ∈ A . for every a ∈ F[s]. (Indeed, assuming that a(0) ̸ = 0, we get that (λa)(0) = λ(0)a(0) ̸ = 0.) So, A is non-catastrophic. Example 2. Let again n = 2, and let A = s1 F[s] + (1 + s2 )F[s]. Every polynomial in A vanishes at (0, −1), and therefore 1∈ / A. We thus have: (1 + s2 ) · 1 ∈ A and

1∈ / A.

So, A is catastrophic. 5. Non-catastrophicity in terms of linear systems For convenience of readers, we briefly recall what are (discrete time) linear systems and how they are connected with convolutional codes. This connection will be derived from the Lefschetz duality as presented in the preliminary section. Remark. The connection between convolutional codes and linear systems (in the classical case) was described first in Rosenthal et al. [16]. Using Oberst’s duality, it was generalized then to the multidimensional setting by Gluesing-Luerssen et al. [5]. A (discrete time) linear system with signal number q is a closed, linear and backward-shift invariant subspace in F[[t ]]q . Remark. The multiplications by s1 , . . . , sn can be interpreted as partial forward-shift operators in F[s], and an analogous definition can be given for convolutional codes: A convolutional code of length q is a closed, linear and forward-shift invariant subspace in F[s]q . Consider the bilinear form

F[s]q × F[[t ]]q → F

(3)

that was introduced in Section 2. Warning. The symbol ‘‘⊥’’ will be used, in this section, to denote the orthogonal complement with respect to (3). Lemma 5. If A is a convolutional code, then A⊥ is a linear system. Conversely, if B is a linear system, then B⊥ is a convolutional code. Proof. This is easy, and is left to the reader.

□

Lemma 6. Let X ⊆ F[s]q and Y ⊆ F[[t ]]q be closed linear subspaces. Then, X ⊥⊥ = X

and

Y ⊥⊥ = Y .

Proof. The proof uses the Hahn–Banach theorem, and is similar to that of Lemma 1. We present the proof of the first relation and leave to the reader to prove the second one. It is clear that X ⊆ X ⊥⊥ . Suppose that there exists x ∈ X ⊥⊥ such that x ∈ / X . Then, by the Hahn–Banach theorem, there is a continuous linear form y on F[s]q such that y|X = 0

and

y(x) ̸ = 0.

We can view y as an element of F[[t ]]q and rewrite the conditions above as y ∈ X⊥

and ⟨x, y⟩ ̸ = 0.

This implies that x ∈ / X ⊥⊥ . The proof is complete. □ As an immediate consequence of Lemma 6 one has the following important fact. Corollary 3. Let A ⊆ F[s]q be a convolutional code and B ⊆ F[[t ]]q a linear system. Then, A⊥⊥ = A

and

B⊥⊥ = B.

As already mentioned, every convolutional code A ⊆ F[s]q has an image representation A = ImG, where G ∈ F[s]qו . The following lemma says that every linear system has a kernel representation.

V. Lomadze / Discrete Mathematics 343 (2020) 111789

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Lemma 7. A linear system B ⊆ F[[t ]]q has a representation B = KerR(σ ), where R ∈ F[s]•×q . Proof. Let A = B⊥ , and let G be a generator matrix of A. Let p be the column number of G, and put R = Gtr . For every x ∈ F[s]p and every y ∈ F[[t ]]q ,

⟨Gx, y⟩ = ⟨Rtr x, y⟩ = (xtr (σ )R(σ )y)(0) = ⟨x, R(σ )y⟩. From this, using non-degenerateness of the bilinear form F[s]p × F[[t ]]p → F, we get that ImG⊥ = KerR(σ ). It follows that B = A⊥ = ImG⊥ = KerR(σ ). The proof is complete.

□ q

Given a subset Y ⊆ F[t ] , let Y denote its closure in F[[t ]]q . We are ready now to formulate and prove the following result. Theorem 2. Let A ⊆ F[s]q be a convolutional code, and let B its dual linear system. Then, A is non-catastrophic if and only if the finite trajectories in B are dense, i.e., B ∩ F[t ]q = B. Proof. The orthogonal complement of A with respect to (1) is B ∩ F[t ]q . The orthogonal complement of this latter with respect to (1) is the same as the one with respect to (3), that is, (B ∩ F[t ]q )⊥ . So, the double orthogonal complement of A with respect to (1) is (B ∩ F[t ]q )⊥ . Next, by Lemma 6, A = B⊥ . The non-catastrophicity condition A = A⊥⊥ , can be rewritten as B⊥ = (B ∩ F[t ]q )⊥ . By the Hahn–Banach theorem, this is equivalent to the relation B = B ∩ F[t ]q . The proof is complete.

□

6. ‘‘Massey-Sain criterion’’ for non-catastrophicity A necessary and sufficient condition for non-catastrophic (classical) convolutional encoders was first obtained by Massey and Sain [13]. They proved that the code generated by a full column rank polynomial matrix G is noncatastrophic if and only if the greatest common divisor of the full-size minors of G is of the form sl for some integer l ≥ 0. In his thesis, Weiner [19] generalized the result of Massey and Sain for multidimensional free convolutional codes. (A convolutional code is said to be free if it is free as a module.) In order to generalize the Massey–Sain criterion to an arbitrary case, let us first reformulate it as follows. (Assume that n = 1) and let A ⊆ F[s]q be a convolutional code with a full column rank generator matrix G. Letting p denote the column number of G, we have an injective homomorphism G

F[s]p → F[s]q . Then, A is non-catastrophic if and only if this homomorphism is still injective modulo every irreducible polynomial λ with nonzero constant term. This is clear in view of the following remark: If λ is an irreducible polynomial, then saying that λ divides all full size minors of G is equivalent to saying that the map Gmodλ

(F[s]/(λ))p → (F[s]/(λ))q is not injective. Let A ⊆ F[s]q be a convolutional code. Define the homological dimension of A to be the minimal number l for which there exists a sequence (Gl , . . . , G1 ) of polynomial matrices such that Gl

G1

0 → F[s]pl → F[s]pl−1 → · · · → F[s]p1 → F[s]q , where p1 , . . . , pl are appropriately defined positive integers, is exact. (An exact sequence is a sequence of modules and homomorphisms between them such that the kernel of one homomorphism equals the image of the next.) The sequence (G1 , . . . , Gl ) is called a polynomial resolution of A and the sequence (p1 , . . . , pl ) the size of this resolution.

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V. Lomadze / Discrete Mathematics 343 (2020) 111789

The homological dimension measures the complexity of a convolutional code and should be an important integer invariant. By Hilbert’s syzygy theorem (see Theorem 4.15 in Lang [9, Ch. XXI], the homological dimension always is less than or equal to n. Notice that free convolutional codes are those ones that have homological dimension 1. Remark. In the 1D case, a polynomial resolution is just a full column rank generator matrix. So, polynomial resolutions can be viewed as generalizations of classical generator matrices. The following theorem can be regarded as a higher dimensional version of the Massey–Sain criterion for noncatastrophicity. Theorem 3. Let A be a convolutional code, and let (G1 , . . . , Gl ) be its polynomial resolution of size (p1 , . . . , pl ), say. Then A is non-catastrophic if and only if the sequence Gl modλ

→ · · · → (F[s]/(λ))p1

0 → (F[s]/(λ))pl

G1 modλ

→ (F[s]/(λ))q

(4)

is exact for every irreducible polynomial λ with nonzero constant term. This is immediate from the following lemma. Lemma 8. Let M be a module over a (commutative) ring and λ a non zero-divisor element of this ring. Suppose that

· · · → L2 → L1 → L0 → M → 0 is an exact sequence with free modules Li . Then, the following two conditions are equivalent: (a) λ is not a zero-divisor on M; (b) the sequence

· · · → L2 /λL2 → L1 /λL1 → L0 /λL0 is exact. Proof. See Corollary 1 in [11]. Example 3. Let n = 2, and consider the code s1 F[s] + s1 (1 + s2 )F[s] ⊆ F[s]. The sequence [

]

− 1 − s2

[

1

0 → F[s]

F[s]

→

2

s1

s1 (1 + s2 )

→

]

F[s]

is exact, and so (

[

s1

s1 (1 + s2 )

]

,

[

−1 − s2 1

] )

is a polynomial resolution of our code. Take any polynomial λ with λ(0) ̸ = 0, and let us check that the sequence [

−1 − s2 modλ 1

0 → F[s]/(λ) is exact.

]

(F[s]/(λ))

→

−1 − s2 modλ

[ 2

s1 modλ

(s1 (1 + s2 ))modλ

→

]

F[s]/(λ)

xmodλ Clearly, is injective. To show exactness at the term (F[s]/(λ)) , assume that lies in the 1 ymodλ [ ] kernel of s1 modλ (s1 (1 + s2 ))modλ . Then, λ divides s1 x + s1 (1 + s2 )y. Since λ is coprime with s1 , it follows that λ divides x + (1 + s2 )y. This is equivalent to saying that

[

]

2

(

)

xmodλ = −(1 + s2 )ymodλ, i.e.,

(

xmodλ ymodλ

)

[ =

−1 − s2 modλ 1

]

ymodλ.

This implies what we want. Remark. A convolutional code A ⊆ F[s]q is called observable (see Gluesing-Luerssen et al. [5]) if it can be described through A = {f ∈ F[s]q | Hf = 0}, where H ∈ F[s]•×q is a polynomial matrix, called a parity check matrix or a syndrome former. The code A is observable if and only if the quotient module F[s]q /A is torsion free. So, the observability property is stronger than non-catastrophicity. With notation of Theorem 3, A is observable if and only if (4) is exact for every irreducible polynomial λ.

V. Lomadze / Discrete Mathematics 343 (2020) 111789

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7. Concluding remark Catastrophic convolutional codes are very much undesirable as a finite number of channel errors may give rise to an infinite number of errors in the decoded sequence. The Massey–Sain classical criterion for non-catastrophicity was generalized by Weiner to multidimensional free convolutional codes. However, in higher dimensions, convolutional codes are not necessarily free and therefore it is not always possible to represent them via full column rank generator matrices. Following one of the main ideas of homological algebra, we propose to represent them via what we call polynomial resolutions. Using these resolutions, we generalize the Massey–Sain criterion to general multidimensional convolutional codes. This generalization is the main result of the paper. Declaration of competing interest The authors declare that there is no conflict of interest in this paper. References [1] N. Bourbaki, Commutative Algebra, Springer-Verlag, New-York, 1989, (Chapters 1–7). [2] C. Charoenlarpnopparut, Applications of Gröbner bases to the structural description and realization of multidimensional convolutional code, Sci. Asia 35 (2009) 95–105. [3] J.-J. Climent, D. Napp Avelli, C. Perea, R. Pinto, MDS 2D convolutional codes, in: 20th International Symposium on Mathematical Theory of Networks and Systems, Melbourne, Australia, 2012, pp. 9–13. [4] E. Fornasini, M.E. Valcher, Algebraic aspects of two-dimensional convolutional codes, IEEE Trans. Inform. Theory 40 (1994) 1068–1082. [5] H. Gluesing-Luerssen, J. Rosenthal, P. Weiner, Duality between multidimensional convolutional codes and systems, in: F. Colonius, U. Helmke, D. Prätzel-Wolters, F. Wirth (Eds.), Advances in Mathematical Systems Theory, Boston, MA, 2001, pp. 135–150. [6] A. Grothendieck, Local Cohomology, in: Lecture Notes in Mathematics, No. 41, Springer-Verlag, New-York, 1967. [7] P. Jangisarakul, C. Charoenlarpnopparut, Algebraic decoder of a multidimensional convolutional code: constructive algorithms for determining syndrome decoder and decoder matrix based on Gröbner basis, Multidimens. Syst. Signal Process. 22 (2011) 67–81. [8] B. Kitchens, Multidimensional convolutional codes, SIAM J. Discrete Math. 15 (2002) 367–381. [9] S. Lang, Algebra, Springer-Verlag, New York, 2002. [10] S. Lefschetz, Applications of Algebraic Topology, Springer-Verlag, New York, 1975. [11] V. Lomadze, The PBH test for multidimensional LTID systems, Automatica 49 (2013) 2933–2937. [12] V. Lomadze, The predictable degree property and minimality in multidimensional convolutional coding, Discrete Math. 342 (2019) 784–792. [13] J.L. Massey, M.K. Sain, Inverses of linear sequential circuits, IEEE Trans. Comput. 17 (1968) 330–337. [14] D. Napp Avelli, C. Perea, R. Pinto, Column distances for 2D-convolutional codes, in: Proceedings of the 19th International Symposium on MTNS, Budapest, Hungary, 2010. [15] D. Napp Avelli, C. Perea, R. Pinto, Input-state-output representations and constructions of finite-support 2D convolutional codes, Adv. Math. Commun. 4 (2010) 533–545. [16] J. Rosenthal, J.M. Schumacher, E.V. York, On behaviors and convolutional codes, IEEE Trans. Inform. Theory 42 (1996) 1881–1891. [17] W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973. [18] M.E. Valcher, E. Fornasini, On 2D finite support convolutional codes: an algebraic approach, Multidimens. Syst. Signal Process. 5 (1994) 231–243. [19] P. Weiner, Multidimensional convolutional codes (Ph.D. dissertation), University of Notre Dame, USA, 1998. [20] E. Zerz, On multidimensional convolutional codes and controllability properties of multidimensional systems over finite rings, Asian J. Control 12 (2010) 117–236.