Two-dimensional convolutional codes are considered, with codewords having compact support indexed in N 2 and taking values in F n , where F is a finite field. Input-state-output representations of these codes are introduced and several aspects of such representations are discussed. Constructive procedures of such codes with a designed distance are also presented.
Maximum Distance Separable (MDS) block codes and MDS one-dimensional (1D) convolutional codes are the most robust codes for error correction within the class of block codes of a fixed rate and 1D convolutional codes of a certain rate and degree, respectively. In this paper we generalize this concept to the class of two-dimensional (2D) convolutional codes. For that we introduce a natural bound on the distance of a 2D convolutional code of rate k/n and degree δ, which generalizes the Singleton bound for block codes and the generalized Singleton bound for 1D convolutional codes. Then we prove the existence of 2D convolutional codes of rate k/n and degree δ that reach such bound when n ≥ kif k | δ, by presenting a concrete constructive procedure.
Index Terms2D convolutional code, generalized Singleton bound, maximum distance separable code, superregular matrix, circulant Cauchy matrix.
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