J. Rosenthal and F. V. York, "BCH convolutional codes," IEEE Transactions on Information Theory, vol. 45, pp. 1833--1844, September 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... codes (see, e.g. 32] and not to fixed ones, those time varying codes were much less utilised in practice than the time invariant counterparts (see, nevertheless, 20] Our approach is another instance of the well known ties between convolutional codes and linear systems (see, e.g. [4, 15, 16, 17, 18, 19, 20, 21, 25, 24, 28, 29, 30]) Our main mathematical tool is a particular module theoretic setting for linear control [5, 7, 8, 11, 14] which has been quite useful in practice (see, e.g. 12, 13] We are utilising some elementary notions of di#erence algebra [2] homological algebra [31] and non commutative algebra [23, ....

J. Rosenthal, E.V. York, BCH convolutional codes, IEEE Trans. Informat. Theory 45 (1999) 1833-1844.

....the minimum of the weights of all the nite codewords v [0;j 1] v 0 ; v 1 ; v j 1 ) resulting from an 4 ROXANA SMARANDACHE information sequence u [0;j] u 0 ; u 1 ; u j ) 6= 0. The limit d r 1 = lim j 1 d r j exists and, if the encoder is noncatastrophic, we have (see [11, 2] for details) d c 0 d c 1 : d c 1 = d free = d r 1 : d r 1 d r 0 : 2.2) In terms of state space descriptions d r 1 is equal to the minimal weight of a nonzero trajectory which starts from and returns to the all zero state. d c 1 is equal to the minimal weight ....

J. Rosenthal and E. V. York. BCH convolutional codes. IEEE Trans. Inform. Theory, 45(6):1833-1844, 1999.

....which dominates the early part of this chapter, the reader is referred to [70, 36, 35, 68, 69, 37] 2.1 Behaviors In this section the notion of dynamical systems and behaviors as well as some related concepts are introduced and brie y discussed. This is a review of the material contained in [58, 61, 71]. De nition 2.1.1 A dynamical system is a triple = T ; A; B ) where T R is the time axis, A is the signal alphabet and B A T is called the behavior. The elements of B are called trajectories. We will work with these abstract notions only long enough to write down an alternate de ....

....to reordering) These row degrees for a minimal encoder are known as the Kronecker indices of the code. 2.2 Realization In this section we will develop rst order representations of convolutional codes as de ned in the previous section. Again this section is a review of previous works including [58, 61, 71]. The following theorem proves the existence of a rst order realization. It should be remarked that the results here are dual statements of those in [37] as observed in [58] Theorem 2.2.1 [58] Let C F n [s] be a rate k=n convolutional code of complexity . Then there exist size ( n k) ....

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J. Rosenthal and E.V. York. BCH convolutional codes. Technical report, University of Notre Dame, Dept. of Mathematics, October 1997. Preprint # 271. Available at http://www.nd.edu/~rosen/preprints.html. To appear in IEEE Trans. Inform. Theory.

....C determined by the i=s=o representation given by [A; B; C; D] has d f s 1: Remark 4. 5 That such a matrix C can be chosen and what in particular is a good choice of C, as well as the fact that this construction can be extended to the case where the base field is F 2 is discussed in detail in [32]. Proof Let w(D) w 0 w 1 D : w fl D 2 C , where w t = Then Delta Delta Delta is an input output sequence of the FSM determined by [A; B; C; D] and in particular (y 0 ; y 1 ; y fl ; u 0 ; u 1 ; u fl ) 19 is in the kernel of (4.22) We will show that ....

.... for constructing convolutional codes [1, 10, 11, 20] however the above construction differs from these in the following ways: ffl knowledge of optimal block codes is not required ffl works for any rate ffl by letting q = 2 , m 2 Z one can extend the construction technique to the binary case [32]. Example 4.7 Let p = 1801, ff = 11 and s = 30. Then g 1;1 (D) 315D 749 75D g 1;2 (D) 935D 104D Gamma340D 559 g 2;1 (D) 825D 418D 20 g 2;2 (D) 1 Gamma 442D 672D g 3;1 (D) 858D Gamma ....

J. Rosenthal and E.V. York. BCH convolutional codes. In preparation. 22

....the jth row distance is d r j = min u 6=0 wt(u [0;j] 1 G r j ) 1.5) The limit d r 1 = lim j 1 d r j exists and one has (see, e.g. 7] for every encoder G(D) the relation d c 0 d c 1 111d c 1 = d free d r 1 111d r 1 d r 0 : 1. 6) In terms of state space descriptions [17], 14] d r 1 is equal to the minimal weight of a nonzero trajectory which starts from and returns to the all zero state. d c 1 is equal to the minimal weight of a nonzero trajectory which starts from and not necessarily returns to the all zero state. Furthermore, if the generator matrix G(D) ....

....trajectory which starts from and returns to the all zero state. d c 1 is equal to the minimal weight of a nonzero trajectory which starts from and not necessarily returns to the all zero state. Furthermore, if the generator matrix G(D) is minimal basic, then d c 1 = d r 1 = d free (see [17], 7] for details) It follows that for a basic encoder the minimal weight codewords are generated by finite information sequences. II. THE GENERALIZED SINGLETON BOUND It is certainly a most natural question to ask how large the distance of a rate k=n code of some bounded degree ffi can be. ....

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J. Rosenthal and E. V. York, "BCH convolutional codes," IEEE Trans. Inform. Theory, vol. 45, pp. 1833--1844, Sept. 1999.

....described by the familiar looking (A; B; C; D) representation: x t 1 = Ax t Bu t y t = Cx t Du t ; x 0 = 0; x fl 1 = 0: 3.3) This system is known as the input state output representation for a convolutional code. It describes the dynamics for a systematic and rational encoder. We refer to [16,18,20] for more details. We say that the matrices (A; B) form a controllable pair if rank Gamma B AB : A ffi Gamma1 B Delta = ffi; and we say that (A; C) form an observable pair if (A t ; C t ) is a controllable pair. Once (A; B) form a controllable pair and (A; C) form an observable ....

....that the matrices (A; B) form a controllable pair if rank Gamma B AB : A ffi Gamma1 B Delta = ffi; and we say that (A; C) form an observable pair if (A t ; C t ) is a controllable pair. Once (A; B) form a controllable pair and (A; C) form an observable pair then it was shown in [16,18,20] that the system (3.3) represents a non catastrophic convolutional code of degree ffi and rate k=n. If one is interested in the construction of convolutional codes with some designed distance there is no limitation if one attempts to construct matrices A; B; C; D, with (A; B) controllable and (A; ....

J. Rosenthal and E.V. York. BCH convolutional codes. IEEE Trans. Inform. Theory, 1999. To appear.

....is d r j = min u [0;j] 6=0 wt(u [0;j] G r j ) 1.5) The limit d r 1 = lim j 1 d r j exists and one has (see e.g. 7] for every encoder G(D) the relation: d c 0 d c 1 : d c 1 = d free d r 1 : d r 1 d r 0 : 1. 6) In terms of state space descriptions [17, 14] d r 1 is equal to the minimal weight of a nonzero trajectory which starts from and returns to the all zero state. d c 1 is equal to the minimal weight of a nonzero trajectory which starts from and not necessarily returns to the all zero state. Furthermore, if the generator matrix G(D) is ....

....trajectory which starts from and returns to the all zero state. d c 1 is equal to the minimal weight of a nonzero trajectory which starts from and not necessarily returns to the all zero state. Furthermore, if the generator matrix G(D) is minimal basic, then d c 1 = d r 1 = d free (see [17, 7] for details) It follows that for a basic encoder the minimal weight codewords are generated by nite information sequences. 2 The generalized Singleton bound It is certainly a most natural question to ask how large the distance of a rate k=n code of some bounded degree can be. McEliece [8] ....

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J. Rosenthal and E. V. York, \BCH convolutional codes," IEEE Trans. Inform. Theory, vol. 45, no. 6, pp. 1833-1844, 1999.

....de nitions given in Section 2 and 3. In Section 5 we will give a de nition for convolutional codes in which it is required that the code words have nite support. Such a de nition was considered by Fornasini and Valcher [48, 5] and by the author in collaboration with Schumacher, Weiner and York [42, 44, 49]. The study of behaviors with nite support has been done earlier in the context of automata theory and we refer to Eilenberg s book [1] We show in Section 5 how this moduletheoretic de nition relates to complete, linear and time invariant behaviors by Pontryagin duality. In Section 6 we will ....

....P (z) are invariants of the behavior. The McMillan degree, the transmission rate and the free distance are then de ned in the same way as for behaviors B F n [ z] 5 The module point of view Fornasini and Valcher [5, 48] and the present author in joint work with Schumacher, Weiner and York [42, 44, 49] proposed a module theoretic approach to convolutional codes. The module point of view simpli es the algebraic treatment of convolutional codes to a large degree, and this simpli cation is probably almost necessary if one wants to study convolutional codes in a multidimensional setting [5, 48, ....

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J. Rosenthal and E. V. York. BCH convolutional codes. IEEE Trans. Inform. Theory, 45(6):1833-1844, 1999.

....Along with n and k, there is a third important parameter of a convolutional code C, called the degree. It is defined as the maximal degree ffi of the k Theta k minors of G(D) Note that equivalent encoding matrices have the same degree and that the degree is an invariant of the code. See [7, 8, 9, 10] for details. Let G(D) be a basic generator matrix for the code C. Let i be the ith row degree of G(D) i.e. i = max j deg g ij (D) In the literature [7] the indices i are also called the constraint length for the ith input of the matrix G(D) and the sum P k i=1 i is sometimes called the ....

....of a nonzero trajectory which starts from and returns to the all zero state. d c 1 is equal to the minimal weight of a nonzero trajectory which starts from and not necessarily returns to the all zero state. Furthermore, if the encoder is non catastrophic, then d c 1 = d r 1 = d free (see [10, 7] for details) It follows that for a non catastrophic encoder the minimal weight codewords are generated by finite information sequences. 2 The generalized Singleton bound Once the row degrees 1 ; k of an encoder G(D) are specified one has a natural upper bound on the free distance of a ....

[Article contains additional citation context not shown here]

J. Rosenthal and E. York, "BCH convolutional codes," IEEE Trans. Inform. Theory, vol. 45, no. 6, pp. 1833--1844, 1999.

.... Gamma1 M) for some T 2 Gl ffi n Gammak (F) S 2 Gl ffi (F) Generalized first order representations as described in the above two theorems are very useful in the design of convolutional codes with large distance and which can be encoded in an efficient manner. We refer the interested reader to [8, 9]. Conclusion The paper did show that multidimensional convolutional codes are powerful encoding devices for the transmission of data over a noisy channel. Since these codes are dual objects to multidimensional systems the algebraic theory of linear systems can be fruitfully applied. Diederich ....

J. Rosenthal and E.V. York. BCH convolutional codes. IEEE Trans. Inform. Theory. To appear.

....described by the familiar looking (A; B; C; D) representation: x t 1 = Ax t Bu t y t = Cx t Du t ; x 0 = 0; x fl 1 = 0: 3.3) This system is known as the input state output representation for a convolutional code. It describes the dynamics for a systematic and rational encoder. We refer to [15, 17, 19] for more details. MAXIMUM DISTANCE SEPARABLE CONVOLUTIONAL CODES 7 We say that the matrices (A; B) form a controllable pair if rank Gamma B AB : A ffi Gamma1 B Delta = ffi; and we say that (A; C) form an observable pair if (A t ; C t ) is a controllable pair. Once (A; B) form ....

....that the matrices (A; B) form a controllable pair if rank Gamma B AB : A ffi Gamma1 B Delta = ffi; and we say that (A; C) form an observable pair if (A t ; C t ) is a controllable pair. Once (A; B) form a controllable pair and (A; C) form an observable pair then it was shown in [15, 17, 19] that the system (3.3) represents a non catastrophic convolutional code of degree ffi and rate k=n. If one is interested in the construction of convolutional codes with some designed distance there is no limitation if one attempts to construct matrices A; B; C; D, with (A; B) controllable and (A; ....

J. Rosenthal and E.V. York. BCH convolutional codes. Technical report, University of Notre Dame, Dept. of Mathematics, October 1997. Preprint # 271. Available at http://www.nd.edu/~rosen/preprints.html.

....algorithms for block codes it is required that the convolutional code has a certain algebraic structure. In this way the algorithm cannot be applied efficiently to arbitrary convolutional codes. On the other hand if the convolutional code is of Reed Solomon type (see [15] or of BCH type (see [16, 21]) then the algorithm is capable of decoding convolutional codes in situations where the Viterbi decoding algorithm would not be feasible because of complexity considerations. The paper is structured as follows: In the next section we summarize some basic notions for block codes and convolutional ....

....describe two variations where we expect the algorithm to perform very efficiently. In Section 6 we will show that the Berlekamp Massey algorithm or any of its recent improvements (see e.g. 3] can be invoked to iteratively decode the Reed Solomon and BCH type convolutional codes as presented in [15, 16, 21]. 2 Convolutional Codes and their State Space Description In this section we will provide a short tutorial on block codes and convolutional codes. More details on our state space approach are given in [11, 13, 15, 16, 21] Comprehensive textbooks on convolutional codes are [7, 8, 12] Let F = F q ....

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J. Rosenthal and E.V. York (Oct. 1997). BCH Convolutional Codes. Tech. rep., University of Notre Dame, Dept. of Mathematics, Preprint # 271. Available at http://www.nd.edu/~rosen/preprints.html.

....ffi is given whose free distance is n(ffi 1) the maximal possible distance of all codes with these parameters. Since the designed distance is maximal we call such a code a maximal distance separable (MDS) convolutional code. More recently the authors of this paper in collaboration with E. York [4, 5, 6, 7] gave for arbitrary rates k=n constructions of convolutional codes with a designed free distance. The techniques employed in these papers were new and they heavily relied on algebraic representations of linear systems. The achieved distances in [4, 6, 7] were approximately k n times the best ....

....this paper in collaboration with E. York [4, 5, 6, 7] gave for arbitrary rates k=n constructions of convolutional codes with a designed free distance. The techniques employed in these papers were new and they heavily relied on algebraic representations of linear systems. The achieved distances in [4, 6, 7] were approximately k n times the best possible free distance found among all convolutional codes of rate k=n and complexity ffi. In particular for high rates the results were near optimal. The authors of this paper showed in [5] that the constructions can be refined in order to achieve better ....

[Article contains additional citation context not shown here]

J. Rosenthal and E.V. York. BCH convolutional codes. Technical report, University of Notre Dame, Dept. of Mathematics, October 1997. Preprint # 271. Available at http://www.nd.edu/~rosen/preprints.html.

.... constructions which did exist were mainly based on some quasi cyclic constructions of block codes as e.g. in [1, 2, 4, 6] or they made use of the parity check matrix H(z) of the convolutional code as e.g. in [8, 15] A survey of some of these results can be found in the monograph of Piret [7] In [12, 14, 16] the first author together with York and Schumacher used some basic ideas in systems theory to derive some convolutional codes of Reed Solomon and BCH type. Different from the previous work in this area convolutional codes were constructed using a first order description of the encoder. It was ....

....author together with York and Schumacher used some basic ideas in systems theory to derive some convolutional codes of Reed Solomon and BCH type. Different from the previous work in this area convolutional codes were constructed using a first order description of the encoder. It was explained in [14] that the algebraic structure of those codes can be used to algebraically decode those codes. The paper is structured as follows: In Section 2 we review some of the results reported in [12, 14, 16] and we give conditions on the first order descriptions leading to desirable Both authors were ....

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J. Rosenthal and E.V. York. BCH convolutional codes. Preprint, October 1997.

....over F p . This is done in a manner quite similar to the classical BCH construction. The main difference is that the parity check matrix (4.6) needs to be extended in a way that preserves the factorization. That this can be done, as well as the types of codes this technique yields, is presented in [32]. V. Conclusions In this paper we studied convolutional codes from a module theoretic point of view and we related our framework to systems theory. We showed that the class of linear behaviors having a kernel representation can be considered as dual to the class of convolutional codes. In our ....

J. Rosenthal and E.V. York. BCH convolutional codes. In Preparation.

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J. Rosenthal and F. V. York, "BCH convolutional codes," IEEE Transactions on Information Theory, vol. 45, pp. 1833--1844, September 1999.

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J. Rosenthal and F. V. York, "BCH convolutional codes," IEEE Transactions on Information Theory, vol. 45, pp. 1833--1844, September 1999.

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