MDS 2D convolutional codes with optimal 1D horizontal projections

Abstract

Two dimensional (2D) convolutional codes is a class of codes that generalizes standard one-dimensional (1D) convolutional codes in order to treat two dimensional data. In this paper we present a novel and concrete construction of 2D convolutional codes with the particular property that their projection onto the horizontal lines yield optimal (in the sense of [2]) 1D convolutional codes with a certain rate and certain Forney indices. Moreover, using this property we show that the proposed constructions are indeed Maximum Distance Separable (MDS), i.e., are 2D convolutional codes having the maximum possible distance among all 2D convolutional codes with the same parameters. The key idea is to use a particular type of superregular matrices to build the generator matrix.

A new rank metric for convolutional codes

“…Note that a block code is a convolutional code with δ = 0. For more general classes of convolutional codes see [4,25] In the sequel, we adopt the notation of McEliece [22, p. 1082] and denote a convolutional code of rate k/n and degree δ an (n, k, δ)-convolutional code.…”

“…The following example illustrates the proprieties mentioned above. [1] α [2] α [3] α [4] α [5] α [6] α [7] α [1] α [2] α [3] α [4] α [5] α [6] α [7] α [8] α [2] α [3] α [4] α [5] α [6] α [7] α [8] α [9] 0 0 0 0 α [0] α [1] α [2] α [3] 0 0 0 0 α [1] α [2] α [3] α [4] 0 0 0 0 α [2] α [3] α [4] α [5] …”

A decoding algorithm for 2D convolutional codes over the erasure channel

“…Two-dimensional (2D) convolutional codes were introduced in 1994 by Fornasini and Valcher [8], while multidimensional convolutional codes in general were first studied in 1998 by Weiner [18]. There are some additional research works dealing with the construction of 2D convolutional codes [1,4,5,15]. The papers [1,4,5] provide constructions for 2D MDS convolutional codes, where in [1] the constructed codes have the additional property that their horizontal projections have optimal free distance with respect to their Forney indices.…”

“…There are some additional research works dealing with the construction of 2D convolutional codes [1,4,5,15]. The papers [1,4,5] provide constructions for 2D MDS convolutional codes, where in [1] the constructed codes have the additional property that their horizontal projections have optimal free distance with respect to their Forney indices. In [15], linear systems are used to construct 2D convolutional codes.…”