J. Rosenthal and R. Smarandache. Construction of convolutional codes using methods from linear systems theory. In Proc. of the 35-th Annual Allerton Conference on Communication, Control, and Computing, pages 953--960, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....6.2 Some Classes of Binary Input State Output Convolutional Codes We will present two ideas for how to select the matrices (A; B; C; D) so as to construct codes with properties desirable for this decoding scheme. BCH type codes over larger elds have already been constructed using these ideas in [61, 62, 59, 71]. We will examine maximum distance separable convolutional codes over larger elds in Chapter 7. Our construction here will focus on permutation matrices and matrices with large order over the binary eld. De nition 6.2.1 Let A be a nonsingular matrix with minimum polynomial p A (s) Then ....

J. Rosenthal and R. Smarandache. Construction of convolutional codes using methods from linear systems theory. In Proc. of the 35-th Annual Allerton Conference on Communication, Control, and Computing, pages 953-960, 1997.

....the connection between m D codes and m D systems. The duality developed by Oberst and sketched in Section 2.6 gives a method for translating and applying results of systems theory to coding theory and vice versa. This program is being actively carried out in the 1 D case (see for instance [55, 44, 43]) ii) Further work on distance and distance bounds for m D convolutional codes should be undertaken. A great deal is known about distance bounds for 1 D convolutional codes (see for instance [21, 2] Also in the 1 D case there are finite time algorithms available for computing the distance of ....

J. Rosenthal and R. Smarandache. Construction of convolutional codes using methods from linear systems theory. In Proc. of the 35-th Annual Allerton Conference on Communication, Control, and Computing, pages 953--960, 1997.

....the code C is RX(z) But with this description, the matrix R becomes the scalar matrix having entries the coe#cients of the polynomials that form a generator matrix for the code C. Although we do not decrease the di#culty of the original problem it is still possible to work with this form. In [8] we worked with this form and we derived the following theorem which did improve the distance in certain situations over previously known ones. Theorem 3.1 Let (P, Q) be in the Kronecker canonical form. Assume n # k and let R be a matrix of size n (# k) If the columns of R form the ....

J. Rosenthal and R. Smarandache. Construction of convolutional codes using methods from linear systems theory. In Proc. of the 35-th Annual Allerton Conference on Communication, Control, and Computing, pages 953--960, 1997.

....constructed above: lim ffi 1 d f (C) d max k n : Hence, for very high rates, the codes constructed above are near maximal. However, we note that very large fields are needed in order to construct these codes. For low rates some constructions were provided by the first author and Smarandache [24] which result in better free distances than k n d max . For the situation of rate 1 n Justesen [8] did construct codes with maximal possible free distance d f = n(ffi 1) All these constructions require large field sizes. It is interesting to observe that the construction which we did provide ....

J. Rosenthal and R. Smarandache. Construction of convolutional codes using methods from linear systems theory. In Proc. of the 35-th Annual Allerton Conference on Communication, Control, and Computing, pages 953--960, 1997.

....constructed above: lim ffi 1 d f (C) d max k n : Hence, for very high rates, the codes constructed above are near maximal. However, we note that very large fields are needed in order to construct these codes. For low rates some constructions were provided by the first author and Smarandache [14] which result in better free distances than k n d max . For the situation of rate 1 n Justesen [3] did construct codes with maximal possible free distance. All those constructions require large field sizes. In the next section, we will give techniques for constructions over arbitrary finite ....

J. Rosenthal and R. Smarandache. Construction of convolutional codes using methods from linear systems theory. In Proc. of the 35-th Annual Allerton Conference on Communication, Control, and Computing, 1997. To appear.

....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 ....

....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 distances also for low rate codes. In this paper we further refine the technique and we show how to rederive the result of Justesen [1] for rate 1=n codes. We want to emphasize that the construction we present here is not just a ....

J. Rosenthal and R. Smarandache. Construction of convolutional codes using methods from linear systems theory. In Proc. of the 35-th Annual Allerton Conference on Communication, Control, and Computing, pages 953--960, 1997.

....above: lim ffi 1 d f (C) d max k n : Hence, for very high rates, the codes constructed above are near maximal. However, we note that very large fields are needed in order to construct these codes. For low rates some constructions were provided by the first author and Smarandache [20] which result in better free distances than k n d max . For the situation of rate 1 n Justesen [6] did construct codes with maximal possible free distance d f = n(ffi 1) All these constructions require large field sizes. It is interesting to observe that the construction which we did ....

J. Rosenthal and R. Smarandache. Construction of convolutional codes using methods from linear systems theory. In Proc. of the 35-th Annual Allerton Conference on Communication, Control, and Computing, pages 953--960, 1997.

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