Feedback equivalence of convolutional codes over finite rings

Abstract The approach to convolutional codes from the linear systems point of view provides us with effective tools in order to construct convolutional codes with adequate properties that let us use them in many applications. In this work, we have generalized feedback equivalence between families of convolutional codes and linear systems over certain rings, and we show that every locally Brunovsky linear system may be considered as a representation of a code under feedback convolutional equivalence.


Introduction
Convolutional codes are a powerful tool that is used to correct digital data. These error correcting codes are applied in numerous situations such as the communications with the deep space, the mobile communications or the hard decision codes. Regarding cybersecurity, convolutional codes are applied in several areas in order to improve the e ciency and security of di erent processes. For instance, these codes are used in cryptography to construct cryptosystems (see [1]). They are also employed in [2] as an alternative scheme that avoids the need for global connection in the modeling of networks by trellis representations. On the other hand, concatenated convolutional codes, developed in [3], have been strongly exploited when it is necessary to transmit and hide sensitive information. Recent advances in parallel and serial concatenated convolutional codes focus on their implementation in the construction of turbo codes.
The connection between linear systems and convolutional codes over nite elds has been studied from di erent points of view depending on the approach to convolutional codes that is being used (see [4]). The available representations let us describe the dynamics of the encoders of the codes or constructing convolutional codes with certain good properties such as observability. Recently, the study of convolutional codes over nite elds has been developed through these relations based on rst order representations and input/state/output (I/S/O) representations and references therein [5][6][7][8].
Although the study of error-correcting codes initially started over nite elds, the research of codes over rings has increased due to the applications and properties of these codes. For example, in [9] an encoder over Z/ Z is developed for decoding MPEG-4 images. Recently, in [10], a steganographic protocol has been performed based on convolutional codes over the ring Z/ Z.
Convolutional codes over rings were introduced by Massey and Mittleholzer in [11] and [12]. They also focused on the study of minimal and systematic encoders over rings. However, convolutional codes over rings do not behave in the same way as convolutional codes over elds because their behavior depends strongly on the structure of the underlying ring. The study of properties, encoders, p-basis and dual convolutional codes over nite rings, has been developed in [13][14][15][16] among others. In [17], a bound on the free distance of convolutional codes over Z p r was developed generalizing the results described in [7]. A construction of nonfree MDS convolutional codes over Z p r is also given in [18] with new upper-bounds on the free distance.
The extension of the duality between linear systems and convolutional codes is not easily generalized to all commutative rings with identity. It depends strongly on the ring and the realization theory that we will want to extend. This paper is based on the generalization given in [19,20] of the theory studied in [21][22][23] about the connection between linear systems (reachable input/state/output representations) and convolutional codes (with nite support). The extension developed in [19,20] let us construct observable families of convolutional codes over a nite ring R. We consider noetherian von Neumann regular rings: that is, nite products of elds. In particular, we suppose that More precisely, these families of convolutional codes over R allow us to construct an algebraic system of simultaneous signal encoding in linear coding networks over the ring R, improving the security of the system. This system lets us send the same message m encoded over the ring R to several receivers and every receiver decodes its message µ j over F j = R/m j where m j is the j-th maximal ideal in the spectrum of R. In this situation, if the messages over F j are shared, it would be possible to create the original message that we assume unique: m = (µ , . . . , µ t ). In this case, a nite base of t communication systems is used, so that the message is encoded over the ring R (see Figure 1). Moreover, a continuous net among receivers is not needed. In cybersecurity framework, this multicast encoding could be applied to e ective coding in the cloud where many agents would use the same cloud resources to di erent purposes and several scripts. ;

Receiver t
On the other hand, feedback equivalence of linear systems over commutative rings is one of the main holistic matters in science and engineering (see [24][25][26]). Two linear systems are feedback equivalent if one can transform one into the other by changes of coordinates together with feedback loops. Regarding equivalence relations in coding literature, due to the fact that a convolutional code can have many encoders, most works on the topic are focused on the study of equivalence relations between encoders of the same convolutional code: that is, the study of the conditions in which two encoding matrices are equivalent, and thus they generate the same code (see [4] for a general overview). In the theory of block codes over nite elds, MacWilliams stated that linear block codes are related by weight-preserving isomorphism (see [27]). However, the weight enumerator does not form a complete invariant under monomial equivalence. In [28], a generalization of the above theory for certain convolutional codes over nite elds is given by making use of classical realization theory (driving representation of the code). They proved that all minimal realizations of a given code are feedback equivalent in the sense that the adjacency matrix of reduced encoders turns into an invariant of the code. Regarding linear codes over nite Frobenius rings, the theorem of monomial equivalence of MacWilliams is extended in [29].
In this paper, we use the study of feedback isomorphisms of linear systems over commutative rings (see [30][31][32]) to de ne a feedback convolutional equivalence for families of convolutional codes over a noetherian von Neumann regular ring R. The key is that the set of Kronecker indices of both objects (partitions of the rank of the state space of the system and the complexity of the code) forms a complete family of invariant elements under feedback isomorphism. Moreover, the relation between locally Brunovsky linear systems and families of convolutional codes over R by I/S/O representations allows us to know how many equivalence classes of families of convolutional codes there are with the same Kronecker's indices; i.e. with the same degree under feedback convolutional equivalence. Note that this feedback equivalence relation is valid in the case of nite elds. This paper is organized as follows: In Section 2 we give the related work with the basic results of linear systems, convolutional codes and the relation between them. In Section 3 we give our main results supported by examples. Finally, we give the conclusions, future work and references.

Preliminaries
The rst part of this section is devoted to basic preliminaries about linear systems over rings and some important properties such as reachability and observability. In the second part, we give a review of convolutional codes over nite elds and their connection with linear systems. Finally, we give an overview of the theory of families of convolutional codes over rings and the I/S/O representation considered in this paper.

. Linear systems and feedback isomorphisms over rings
Let R be a commutative ring with identity. A time-invariant linear system Σ = (A, B, C, D) ∈ R δ×δ × R δ×k × R p×δ × R p×k is described as follows where x t ∈ R δ is the state vector, u t ∈ R k is the control vector, and y t ∈ R p is the output vector for each time instant t. The dimension of the state space δ is known as the dimension of the linear system. The generalization of linear systems over a commutative ring with identity R starts from considering a triple Σ = (X, f , B) where X is an R-module of rank δ (state space of the system), f : X → X is an endomorphism of X, and B is a nitely generated submodule of X (see [26]). When X R δ , then a pair of matrices (A, B) ∈ R δ×δ × R δ×k gives a linear map as the one described above by the assignment and thus, we can consider the dynamical system Σ = (A, B) over R.
We review some results about reachability (controllability from the origin) properties of linear systems over R. In the case of time-invariant systems, the reachability of a system refers to the ability of the system to reach x from the origin in some nite time. We review the main characterization of the reachability of a time-invariant linear system in terms of the pair (A, B). [25] and [33]). Let Σ be a linear system over R. The following statements are equivalent 1) Σ is reachable. 2 3) The map ϕ: R kδ → R δ given by multiplication by Φ δ is residually surjective at each maximal ideal m of R.

is surjective (generalization of Hautus Test).
State observability is the ability to determine the state vector of the system knowing the input and the corresponding output over some nite time interval. The following result describes observability properties in terms of the pair of matrices (A, C) of the system. Proposition 2.2 (c.f. Theorem 2.6, [25]). Let Σ be a linear system over R. The following statements are equivalent 1) Σ is observable.
is the ideal of R generated by the δ × δ minors of Ω δ , then, the annihilator of U δ (Ω δ ) is zero.
On the other hand, it is known that two linear systems over a eld K are feedback equivalent when they have a similar feedback dynamical behavior (they have the same feedback invariant elements). By [34] and [35], feedback classi cation of dynamical linear systems over a eld K becomes the feedack classi cation of reachable systems and thus, we can compute the number of feedback equivalence classes of dynamical linear systems over a δ-dimensional K-vectorial space, fe K (δ), by the classi cation of reachable systems. The number of feedback classes of δ-dimensional reachable systems over K may be computed by using the indices (ξ , ξ , . . . , ξp) where The set of indices described in Equation (2) is a complete family of invariants of reachable linear systems under the group of feedback transformations. If we reorder them, then they are equal to the Kronecker's minimal column indices for the matrix pencil (zI − A, B) associated to Σ. We denote them by K Σ = (K , . . . , Kp). The set K Σ is called the set of control invariants of Σ. Moreover, Kronecker indices (K ≥ K ≥ · · · ≥ Kp) form an ordered partition of the integer δ, the dimension of the system; that is, δ = K + . . . + Kp. And then, fe K (δ) is equal to the number of partitions of the integer δ, p N (δ). We recall that both sets of indices {ξ i } and {K i } are conjugated with each other (see [35]). Note that computing the Kronecker's indices of a dynamical linear system by the computation of the conjugated invariant elements is, in some cases, easier than the direct computation. We can use Young's diagram. Example 2.3. Let Σ be the following linear system over Z/ Z: We compute the invariant indices ξ and ξ Therefore, ξ Σ = ( , ). We can represent these indices by the Young's diagram and thus, we transpose the diagram in order to compute the Kronecker indices, then K Σ = (κ , κ ) = ( , ), see Table 1.
The existence of feedback invariants for Σ is studied in [30][31][32], for a speci c type of systems: locally Brunovsky linear systems (systems having locally a Brunovsky Canonical Form). In this case, the invariant modules which form a complete family of invariant elements. Moreover, in [32] it is proved that the classes of feedback isomorphisms of locally Brunovsky linear systems with state space X, a projective nitely generated R-module of constant rank δ, is in bijective correspondence with the set of solutions of the following linear equation in P(R), the monoid of isomorphism classes of nitely generated projective R-modules. Therefore we can compute the number of classes of feedback isomorphisms of locally Brunovsky linear systems with state space X, fe R (m) (where m denotes the class of isomorphisms of X in P(R)) by If the monoid is cancellative, the invariant modules In the case of linear systems over commutative rings, we recall that we can obtain another set of invariants [32]). If R is a von Neumann regular ring, we also have available a characterization of the ring R in terms of reachability and locally Brunovksy properties of linear systems over the ring. Theorem 2.6 (cf. Theorem 3.2 in [36]). Let R be a commutative ring with identity. Let Σ be a linear system over R where X is a nitely generated R-module. Then the following are equivalent i) R is a von Neumann regular ring. ii) Σ is reachable if and only if Σ is a locally Brunovsky linear system.

. Convolutional codes and linear systems
In literature regarding codes, one can nd di erent de nitions for a convolutional code over a nite eld that are related, in di erent ways, with linear systems, see [4]. We have available the linear algebra point of view in which a convolutional code is a linear subspace and its associated linear system was given by Massey and Sain (see [37] and [38]). It is known as driving input/output representation. Another approach to convolutional codes over nite elds is the symbolic dynamics point of view. In this case, a convolutional code is a linear compact irreducible and shift invariant subset of F n [z, z − ] and the relation with linear systems is given in [39], among others. Also, convolutional codes can be de ned as a class of time-invariant and complete behavior in the sense of Willems [40], and there is a realization theory too (see [41,42]). We are interested in the module-theoretical approach of convolutional codes that requires that the codewords have nite support (see [43]). We consider that a rate k n -convolutional code C of degree δ over a nite eld is a free submodule of F[z] n of rank k. In the following, we use the notation of McEliece and we say that C is an (n, k, δ)-convolutional code (see [37]). With this point of view a convolutional code C can be described by an I/S/O representation Σ C = (A, B, C, D); i.e. a reachable linear system over the nite eld. Moreover, if we consider a reachable and observable linear system Σ over a nite eld, then we obtain an observable convolutional code that is usually denoted by C (A, B, C, D) (see [21,22]). In the sequel, we denote it by C(Σ).
The above relation is extended to noetherian von Neumann regular rings in [19,20]. We are going to recall the main results of this generalization: An (n, k)-family of convolutional codes over a commutative ring R, C, is a free submodule C ⊂ R[z] n of rank k, and such that R[z] n /C is at over R. In this case, we can consider families of convolutional codes, one of each prime ideal p in Spec(R). The above de nition allows us to understand a convolutional code over R as a family of convolutional codes parametrized by Spec(R): that is, C over a ring R give rise to a convolutional code over every residue eld of R. Moreover, we have that an encoder for an (n, k) -family of convolutional codes C over R is a matrix such that Im G(z) = C and G(z) is injective.

Remark 2.8. If we can consider a systematic encoder for a family of convolutional codes C, then all encoder of the family of codes will be systematic and thus, we say that C is a systematic family of convolutional codes.
Let C ⊂ R[z] n be an (n, k)-family of convolutional codes over R and let p ∈ Spec R be a prime ideal. Since R[z] n /C is R-at, the reduction of C modulo p, C(p) := C/pC, is an (n, k)-convolutional code over k(p) := Rp/pRp. For each p, we denote by δ(p) the degree of C(p). Thus, in this setting the degree of the family C is no longer an integer but a function δ : Spec(R) → N.

De nition 2.9. Fix δ ∈ N. A family of convolutional codes over R, C, is said to be of degree δ if its degree function (4) is constant and equal to δ.
In the sequel, we focus on an (n, k, δ) -systematic family of convolutional codes, C, over a noetherian von Neumann regular ring R. Since R is zero dimensional and noetherian, Spec(R) is a nite set of prime ideals. Moreover, every prime ideal is maximal and then Spec(R) = {m , . . . , m t }. We denote F j := R/m j . Thus we can consider an (n, k, δ) convolutional code, C j , over each F j . Remark 2.10.

The condition imposed over Coker(G(z)) of being R-at can be translated into some algebraic conditions on the entries of G(z). However, in case R is a von Neumann regular ring these conditions are super uous, since every R-module is at, ie., in this case, any encoder G(z) satis es this condition. 2. Note that if G(z) is a systematic encoder of C in R, then the encoders of each
The inverse does not hold true, in general.
By [19,20] there exists a unique triple of matrices (K, L, M) ∈ R (δ+n−k)×δ × R (δ+n−k)×δ × R (δ+n−k)×n de ning a rst order representation of the code such that and satisfying that the code is described by Ker(zK + L|M) C. Moreover, the above matrices satisfy minimality conditions: 1. K has column full size rank.

(K | M) has row full size rank. 3. Map (zK + L | M) de ned by (zK + L | M) : R[z] δ+n → R[z] δ+n−k is surjective.
From the above representation, we also have available an input/state/ouput (I/S/O) representation that describes the code as a reachable linear system: We can make elementary operations over (K, L, M) and obtain (K, L, M) such that where Σ C ∈ R δ×δ × R δ×k × R (n−k)×δ × R (n−k)×k , and it veri es that The matrices A, B, C and D over R de ne a representation of the code as a controllability state spaces linear system from Equation (5) to where x t ∈ R δ is the state vector, u t ∈ R k is the information vector, y t ∈ R p is the parity vector and v t is a codeword of C for each time instant t. We assume that v t is a nite-weight codeword and the code sequence has nite weight. Note that this I/S/O representation is di erent from the driving representation given in [38].
Notation. If we denote by I j the ideal generated by all components in which R decomposes except F j ; that is, In a similar way, we can obtain Σ C over R by patching Σ Cj , the I/S/O representations of the convolutional codes C j over each F j .
Regarding control properties, by Proposition 2.1 the minimality condition ) of the rst order representation of a family of convolutional codes is actually equivalent to say that Σ C is a reachable linear system. Remark 2.12. By [21] every Σ j is a reachable linear system over F j . Thus, Σ over R is residually surjective. By Proposition 2.1, Σ is reachable over R.
In addition, if Σ C = (A, B) C is the dynamical part of an I/S/O representation of a family of convolutional codes, C, over R, then Σ C is a locally Brunovsky linear system over R by [36]. So it veri es all properties of feedback isomorphism of linear systems described in Subsection 2.1.
An essential property of convolutional codes over nite elds constructed by I/S/O representations as described in Equation (7)  Note that the above property is equivalent to say that a convolutional code C is observable if the quotient F[z] n /C is a at F[z]-module. Lemma 2.11 in [22] ensures that C(Σ) is observable (non-catastrophic convolutional encoder) if and only if Σ C is an observable linear system. Note that Σ is also reachable by minimality conditions. Moreover, the generalization of the above lemma is given in [19] and [20] and it is as follows: if we consider a reachable and observable linear system Σ over R, then C(Σ) is an observable family of convolutional codes in the sense that R[z] n /C is at over R[z].

Feedback convolutional equivalence over noetherian von Neumann regular rings
Before de ning the feedback convolutional equivalence between families of convolutional codes, we recall some results above the degree and the controllability indices of convolutional codes over nite elds (see [23]). These results will be used in the sequel: i) The degree of the encoder G(z) of a convolutional code C over a nite eld, that is denoted by δ(G(z)), is de ned by their column degrees, ν i := max{deg(g i,j (z) where i = , . . . , n}, as δ(G(z)) := k l= ν l . ii) The complexity of a convolutional code C, denoted by δ(C), is de ned as the highest degree of the full size minors of any encoder G(z). iii) An encoder matrix G(z) of C is minimal if and only if δ(G(z)) = δ(C). iv) The set of column degrees of any minimal encoder of C are known as the Forney or controllability indices of the code. We can reorder them if it is necessary such that κ ≥ . . . ≥ κ k . The invariant δ = k i= κ i is the degree of the code C. The complexity of a convolutional code over a nite eld equals its degree. In the following, we denote the set of controllability indices by K C = (κ , ..., κ k ) . v) Note that the controllability indices of a convolutional code are unique and invariants of the code. Since by rst order representation we can compute an I/S/O representation for a convolutional code C ( [21][22][23]), the relation between controllability indices of the code and controllability (reachability) indices of the I/S/O representation associated as dynamical linear system is clear and it is given in the following theorem: We can compute a minimal rst order representation of C; that is, the matrices K, L and M that characterize the encoder G(z): From the above triple of matrices, we compute its I/S/O representation Σ C that is equal to the system Σ obtained in the Example 2.3. Thus K Σ C = (κ , κ ) = ( , ).
Let R be a noetherian von Neumann regular ring. Let C ⊕ t j= C j be an (n, k, δ) systematic family of convolutional codes over R where C j is an (n, k, δ) convolutional code over each R/m j = F j for each maximal ideal m j ∈ Spec(R).
Moreover, note that Σ C Z/ Z described by has Kronecker's indices Example 3.3) and and then ξ = rk(B ) = and ξ = rk B A B − rk(B ) = − = . So, by Table 2, where f .i denotes the feedback isomorphism between locally Brunovsky linear systems with state space of rank δ and Σ C = (A, B) C (respectively for Σ C ).
Proof. An I/S/O representation of a family of convolutional codes over R is a reachable linear system, thus R is a Locally Brunovsky ring by [36]. Therefore, we can apply the results of feedback classi cation of locally Brunovsky linear systems described in Subsection 2.1. Since P(R) = N t is cancellative, the invariants I i classify too, and thus, From P(R) P(F ) × . . . × P(F t ), Equation (8) implies that Now, Σ Cj and Σ Cj are I/S/O representations (reachable dynamical linear systems) over F j . The set of nitely generated projective modules of a nite eld is the cancellative monoid P(F j ) = N. So, Then, by [44], By Lemma 3.9, # { feedback classes of C} = p(δ) t and since R is a noetherian von Neuman regular ring, this number is equal to fe R (δ), so we conclude the proof.
Note that Kronecker indices of a linear system Σ are obtained from the pair (A, B) and hence, they are independent of C and D. Therefore, since controllability invariant indices of an (n, k, δ) -family of convolutional codes, C, are obtained from its I/S/O representation Σ C , it follows that controllability indices of C are obtained from pair (A, B) C , and they are independent of (C, D) C . We highlight this property in the following example. The following system is Σ : Then, we obtain the following minimal rst order representation