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Abstract

ABSTRACT:Over the past few years, multidimensional convolutional code has become an emerging area of research in the signal processing community. While one-dimensional convolutional code and its variants have been thoroughly understood, the m-D counterpart still lacks unified notation and efficient encoding/decoding implementation. Here, the strong link between the theory of Gröbner bases and m-D convolutional code is explored. Several applications of Gröbner bases to the characterization of m-D convolutional encoders are proposed. Furthermore, the more practical problem of minimal encoder realization is discussed and an algebraic algorithm based on the use of Gröbner bases is provided. From the implementation point of view, the syndrome decoder is currently the only means for decoding m-D convolutional code. A constructive method for computing the syndrome decoding matrix using the theory of syzygy modules is proposed.

Optimal Construction for Decoding 2D Convolutional Codes over an Erasure Channel

“…then E is the support of its coefficients with erasures and E h ⊆ E is the support of the coefficients with erasures that are distributed on a horizontal line in N 2 0 in a window W of length t + 1 such that E h ⊂ W. Consider H and v as in (4), as well as H W and v W as in (6). Then, define the vector v Ω by selecting the coefficients v(c, d) of v with (c, d) ∈ Ω ν (W)\W.…”

“…In [3], the authors defined the free distance of a 2D convolutional code, established an upper bound for this distance, and then presented some constructions of the 2D convolutional codes with an optimal free distance. A generalization of these codes, called nD convolutional codes, were first introduced in [4,5] and then further developed in [6][7][8]. However, decoding these kind of codes is a barely explored topic, which we will address in this paper.…”

Burst Erasure Correction of 2D Convolutional Codes

“…If the code C admits a right factor prime encoder [3], then it can be equivalently described using an (n − k) × n full rank polynomial matrix H(z 1 , z 2 ), called paritycheck matrix of C , as…”