COMPOSITION CODES

. In this paper we introduce a special class of 2D convolutional codes, called composition codes, which admit encoders G ( d 1 ,d 2 ) that can be decomposed as the product of two 1D encoders, i.e., G ( d 1 ,d 2 ) = G 2 ( d 2 ) G 1 ( d 1 ). Taking into account this decomposition, we obtain syndrome formers of the code directly from G 1 ( d 1 ) and G 2 ( d 2 ), in case G 1 ( d 1 ) and G 2 ( d 2 ) are right prime. Moreover we consider 2D state-space realizations by means of a separable Roesser model of the encoders and syndrome formers of a composition code and we investigate the minimality of such realizations. In particular, we obtain minimal realizations for composition codes which admit an encoder G ( d 1 ,d 2 ) = G 2 ( d 2 ) G 1 ( d 1 ) with G 2 ( d 2 ) a systematic 1D encoder. Finally, we investigate the minimality of 2D separable Roesser state-space realizations for syndrome formers of these codes. that the special structure of composition codes can be exploited to construct 2D convolutional codes with good distance properties based on 1D results. Moreover, we think that it will allow developing a decoding algo-rithm based on a sequencial application of 1D decoding procedures. This would be a great advantage since there are no decoding algorithms for 2D convolutional codes.


Introduction
In this paper we define a new class of two-dimensional (2D) convolutional codes, called composition codes. These codes admit encoders G(d 1 , d 2 ) that can be obtained from the series connection of two one-dimensional (1D) encoders G 1 (d 1 ) and G 2 (d 2 ), i.e., as G(d . This decomposition allows us to apply the well-developed theory of 1D convolutional codes to the study of composition codes. It is our conviction that the special structure of composition codes can be exploited to construct 2D convolutional codes with good distance properties based on 1D results. Moreover, we think that it will allow developing a decoding algorithm based on a sequencial application of 1D decoding procedures. This would be a great advantage since there are no decoding algorithms for 2D convolutional codes. Restricting to these codes we first obtain syndrome formers directly from G 1 (d 1 ) and G 2 (d 2 ). We then focus on state-space realizations of encoders and syndrome formers of composition codes. For that, we consider 2D state-space realizations by means of separable Roesser models. Such models admit a characterization of minimality of the dimension of realizations in terms of the corresponding matrices, which does not happen if we consider other models [1,2,12]. Moreover, the problem of 2D state-space realization by means of separable Roesser models can be reduced to two sequential 1D realization problems.
We investigate how to obtain minimal state-space realizations (realizations with minimal dimension) of composition codes for code generation and code verification. This question has been solved for 1D convolutional codes [3,5,6] by means of characterizing the encoders and syndrome formers with realizations of minimal dimension among all the encoders and syndrome formers of the code, respectively. These encoders and syndrome formers are called minimal. However, the characterization of minimal encoders and syndrome formers is still an open problem for the 2D case.
A characterization of minimal encoders for general 2D convolutional codes of rate 1/n was obtained in [9,10]. However, the generalization of the results in [9,10] for 2D convolutional codes of rate k/n, with k > 1, appears to be very di cult. However the problem becomes easier to handle if we restrict our study to the classes of composition codes, and in particular to those which admit an encoder is systematic. Here, we obtain minimal encoders for such codes and study the minimality of realizations of their syndrome formers.
This paper is organized as follows. In the next section we present some preliminaries on 1D and 2D polynomial matrices. In section 3 we give the basic notions on 2D convolutional codes. In section 4 we introduce the composition codes and give a construction of syndrome formers for composition codes which admit encoders are right prime. Such construction is obtained from G 1 (d 1 ) and G 2 (d 2 ). State-space realizations by means of separable Roesser models of encoders and syndrome formers of a 2D convolutional code are presented in section 5. Finally, in section 6 composition codes which admit is systematic are considered and minimal encoders of such codes are obtained. Minimal syndrome formers among a class of syndrome formers of such codes are also obtained.

Preliminaries
In this paper we adopt the usual notation of F • unimodular if n = k and it has polynomial inverse; • right prime if it has full column rank and for every factorization All statements on "column" and "right" factors can be couched in "row" and "left" terms, upon taking transposes. Let The maximal order minor of G(d) constituted by the rows 1  r 1 < r 2 < · · · < r k  n and the maximal order minor of H(d) constituted by the rows 1  s 1 < s 2 < · · · < s n k  n are said to be corresponding if {r 1 with columns degrees`1,`2, . . . ,`k.
• The external degree of G(d), extdeg(G), is the sum of its column degrees, i.e., extdeg(G) = P k i=1`i ; • The internal degree of of G(d), intdeg(G), is the maximum degree of its full size minors.
Next we consider 2D polynomial matrices. Concerning matrix factorization, there exist two notions of primeness for such matrices.
is: • unimodular if n = k and it has polynomial inverse; • right factor prime (rFP) if it has full column rank and for every factorization is unimodular; • right zero prime (rZP) if it has full column rank and the ideal generated by the maximal order minors of

2D convolutional codes
In this paper we consider 2D convolutional codes constituted by sequences indexed by Z 2 and taking values in F n , where F is a field. Such sequences {w(i, j)} (i,j)2Z 2 can be represented by bilateral formal power serieŝ In the sequel we shall use the sequence and the corresponding series interchangeably, depending on the problem we are dealing with. For n 2 N, the set of bilateral formal power series over F n is denoted by F n 2D . This set is a module over the ring It follows, as a consequence of [Theorem 2.2, [7]], that a 2D convolutional code can always be given as the image of a full column rank polynomial operator G(d . Such polynomial matrix is called an encoder of C. A code with encoders of size n ⇥ k is said to have rate k/n.
Note that this definition of code di↵ers from the definition in [13,14], where only finite support codewords are considered. Moreover it also di↵ers from the one in [4] where non full column rank 2D polynomial matrices are allowed as encoders. However, our definition is motivated by the fact that only full column rank encoders are relevant for the purpose of obtaining minimal realizations of a code.
Two encoders, , the convolutional encoders are unique up to the post-multiplication by a square nonsingular 2D rational matrix. If are both right factor prime then Thus, if C admits a rZP encoder, then all its rFP encoders are rZP.
A 2D convolutional code C of rate k/n can also be represented as the kernel of a (n k) ⇥ n left factor prime polynomial matrix.
Given a right factor prime encoder of C, a syndrome former of C can be obtained by constructing a (n k) ⇥ n left-factor prime matrix is unimodular. This means that if a 2D convolutional code C admits a rZP encoder, then the corresponding syndrome formers are lZP (see Proposition 2.5).
1D convolutional codes and its encoders and their syndrome formers are defined in a similar way as for the 2D convolutional codes, but are in this case polynomial matrices in one indeterminate d (instead of d 1 and d 2 ) [3, 5, 6].

Composition codes
In this section we consider a particular class of 2D convolutional codes generated by 2D polynomial encoders that are obtained from the composition of two 1D polynomial encoders. Such encoders/codes will be called composition encoders/codes. The formal definition of composition encoders is as follows.
are 1D encoders, is said to be a composition encoder.
Note that the requirement that G i (d i ), for i = 1, 2, is a 1D encoder is equivalent to the condition that G i (d i ) is a full column rank matrix. Moreover this requirement clearly implies that G 2 (d 2 )G 1 (d 1 ) has full column rank, hence the composition G 2 G 1 of two 1D encoders is indeed a 2D encoder.
The 2D code C associated with G = G 2 G 1 , given as can be factorized as follows: , with N 2 and N 1 constant matrices. If N 2 has full column rank and N 1 has full row rank we say that (1) is an optimal decomposition of M (d is an optimal decomposition of a composition encoder G(d 1 , d 2 ), then G 2 (d 2 ) and G 1 (d 1 ) are full column rank matrices.
In the sequel we shall focus on the syndrome formers of composition codes. Since composition encoders can be written as a product of two 1D convolutional encoders, we use this property for constructing syndrome formers of the corresponding code based on 1D polynomial methods. For that purpose we shall concentrate on composition codes that admit an encoder is rZP and therefore all rFP encoders of the code are rZP. Moreover, the corresponding syndrome formers are also lZP (see Proposition 2.5). Since is left prime and is such that This reasoning leads to the following proposition.
as in (2). Then is a syndrome former of Proof. Since (3) constitutes a polynomial right inverse of H(d 1 , d 2 ). Consequently H(d 1 , d 2 ) is left zero prime which implies that it is left factor prime as we wish to prove.

State-space realizations of encoders and syndrome formers
In this section we recall some fundamental concepts concerning 1D and 2D statespace realizations of transfer functions, having in mind the realizations of encoders and syndrome formers.
A 1D state-space model ( On the other hand, when considering a realization ⌃ 1D (A, B, C, D) of a syndrome former H(d), the codewords w are the inputs u that yield zero output.
The encoders and syndrome formers of a 1D convolutional code C with minimal McMillan degree among all the encoders and all syndrome formers of C, respectively, are said to be minimal. Minimal encoders and minimal syndrome formers of C have McMillan degree equal to the degree of the code C, where the degree of C is defined as the external degree of the right prime and column reduced encoder of the code. Such encoders are called canonical and constitute a particular class of minimal encoders [3,5].
Minimal encoders and syndrome formers of a 1D convolutional code were completely characterized in [3,5,6]. Such characterizations are given in terms of the properties of the encoders and syndrome formers as polynomial matrices. Another characterization of minimal encoders is given by the following theorem. Considering the 2D case, there exist several types of state-space models [1,2]. In our study we shall consider separable Roesser models [12]. These models have the following form: (4) 8 > < > : , C 2 and D are matrices over F, with suitable dimensions, u is the input-variable, y is the output-variable, and x = (x 1 , x 2 ) is the state variable, where x 1 and x 2 are the horizontal and the vertical state-variables, respectively. The dimension of the system described by (4) . Then the 2D system obtained as the series concatenation of these two realizations (by considering u 2 (i, j) := y 1 (i, j)) is a realization of M (d 1 , d 2 ) given by 8 > > > < > > > : > : As shown in [8,9] . Note that, since both encoders and syndrome formers are (2D) polynomial matrices, they both can be realized by means of (4). However, as already mentioned, when considering realizations of an encoder G(d we shall take A 12 = 0 and y = w; on the other hand when considering realizations of a syndrome former H(d , we shall take A 21 = 0, u = w and y = 0. This means that when considering encoders we are interested in the input/output behavior of a 2D state-space model of the form (4), with A 12 = 0, whereas when we consider syndrome formers we are interested in the output-nulling inputs of a 2D state-space model of the form (4), with A 21 = 0. As happens in the 1D case, we say that an encoder and a syndrome former of a 2D convolutional code C are minimal if they have minimal Roesser McMillan degree among all encoders and syndrome formers of C, respectively. However, contrary to what happens in the 1D case it seems hard to obtain a characterization for minimal encoders and for minimal syndrome formers. In [11] su cient conditions were established that guarantee the minimality of an encoder of a code. These su cient conditions are given in the following result. Theorem 5.3. Let C be a 2D convolutional code and G(d satisfy the conditions of Theorem 5.1. Then G(d 1 , d 2 ) is a minimal encoder of C. As we shall see, the question of minimal realization seems less hard to handle for composition codes.

Minimal realizations of composition codes
In this section we restrict our study to 2D composition encoders that admit a special structure, namely, in which G(d is a systematic encoder. where T 2 F n⇥n is an invertible constant matrix andḠ(d) 2 F (n k)⇥k [d].
Example 6.2. In Z 2 , consider the polynomial encoder given by invertible andḠ(d) = Note that this definition is slightly di↵erent from the usual one (see for instance [3]) as T is any invertible matrix rather than a permutation matrix.
systematic then it is a minimal encoder of C = Im G(d).
Let C be a composition code generated by a composition encoder G(d , for some p 2 N, is a systematic encoder, and is a minimal encoder. Note that the minimality assumption on G 1 (d 1 ) is not restrictive. In fact, we can assume without loss of generality that G 1 (d 1 ) is right prime, and in case G is a minimal encoder of the corresponding 1D convolutional code [3,5], and moreover, is also an encoder of C. Let ⌃ 1D (A 11 , B 1 ,C 1 ,D 1 ) and ⌃ 1D (A 22 ,B 2 , C 2 ,D 2 ) be minimal realizations of G 1 (d 1 ) and G 2 (d 2 ), respectively. Observe that, since G 1 (d 1 ) is a minimal encoder of the 1D code C 1 = Im G 1 (d 1 ) and G 2 (d 2 ) is a minimal encoder of the 1D code C 2 = Im G 2 (d 2 ) (see Proposition 6.3), it follows that the realizations ⌃ 1D (A 11 , B 1 ,C 1 ,D 1 ) and ⌃ 1D (A 22 ,B 2 , C 2 ,D 2 ) satisfy Theorem 5.1. As already shown, connecting in series ⌃ 1D (A 11 , B 1 ,C 1 ,D 1 ) and ⌃ 1D (A 22 ,B 2 , C 2 ,D 2 ) yields the following 2D realization of G(d 1 , d 2 ): 8 > < > : . The next theorem shows that, under the technical condition that ⇥C given by (7) is a minimal realization of the composition code C.
be a composition encoder of a 2D convolutional code C, such that , for some p 2 N, is a minimal 1D encoder. Moreover, let ⌃ 1D (A 11 , B 1 ,C 1 ,D 1 ) and ⌃ 1D (A 22 ,B 2 , C 2 ,D 2 ) be two 1D minimal realizations of G 2 (d 2 ) and G 1 (d 1 ), respectively, and assume that ⇥C 1D1 ⇤ is square and invertible. Then , respectively, it follows, by Theorem 5.1 that they satisfy the following conditions.  (ii) follows immediately from Condition 2.
(iii) Note that sinceB 2 = L 2D2 andD 2 has full column rank, has also full column rank. Then there exists a matrix U such that On the other hand, Condition 3 implies that there exists a matrix L 1 such that B 1 = L 1D1 . Then due to (8) and to the fact that ⇤ 1 is, by definition, a mla ofD 1 . This implies that a mla of F can be obtained by (if necessary) adding extra rows to ⇤ 1 U .

Let then⇤
, for a suitable matrix T , be a mla of F . Now, the are 1D syndrome formers of the 1D convolutional codes Im G 1 (d 1 ) and Im G 2 (d 2 ), respectively. Let It is easy to see that H(d 1 , d 2 ) is a syndrome former of C. Moreover, it can be shown that it is possible to assume, without loss of generality, that (10) is an optimal decomposition of H(d 1 , d 2 ). Therefore: is a syndrome former of the 1D convolutional code Im G 1 (d 1 ) and G 1 (d 1 ) is a minimal encoder of Im G 1 (d 1 ), it follows that µ(L 1 ) µ(G 1 ), [3,5,6], and hence µ R (H) µ R (G). Further, µ( Proof. Assume that X(d 2 )G 1 (d 1 ) = 0 and write X(d 2 ) = X i 0 X i d i 2 , X i 2 F`⇥ p . Then for all i 0, X i G 1 (d 1 ) = 0. Since X i is a matrix over F and G 1 (d 1 ) has full row rank over F, this means that X i = 0, for all i 0, and therefore X(d 2 ) is a null polynomial matrix. Theorem 6.7. Let C = Im G(d 1 , d 2 ) be a 2D composition code of the form (6), where G(d 1 ) has full row rank over F. Let furtherH(d 1 , T be a syndrome former of C, where X 1 (d , . Then µ R (H) µ R (G).
Proof. Note thatH(d 1 , d 2 )G(d 1 , d 2 ) = 0 if and only if ( must be a syndrome former of the 1D convolutional code Im G 1 (d 1 ) and consequently µ(X 1 ) µ(G 1 ), [6]. On the other hand, since by assumption G 1 (d 1 ) has full row rank over F, by Lemma 6.6, we have that X 21 (d 2 ) + X 22 (d 2 )Ḡ 2 (d 2 ) = 0, which is equivalent to = 0, and . Hence µ ⇥ X is a minimal encoder of Im . Now, sinceH(d T , it is not di cult to see that Corollary 1. Using the notation and conditions of Theorem 6.7, the syndrome former of C given by (10) has minimal Roesser McMillan degree among all syndrome formers of the same structure.