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The notion of Generalized Hamming weights of block codes has been investigated since the nineties due to its significant role in coding theory and cryptography. In this paper we extend this concept to the context of convolutional codes. In particular, we focus on column distances and introduce the novel notion of generalized column distances (GCD). We first show that the hierarchy of GCD is strictly increasing. We then provide characterizations of such distances in terms of the truncated parity-check matrix of the code, that will allow us to determine their values. Finally, the case in which the parity-check matrix is in systematic form is treated. [ABSTRACT FROM AUTHOR]

A convolutional code C over Z p r [ D ] is a Z p r [ D ] -submodule of Z p r n [ D ] where Z p r [ D ] stands for the ring of polynomials with coefficients in Z p r . In this paper, we study the list decoding problem of these codes when the transmission is performed over an erasure channel, that is, we study how much information one can recover from a codeword w ∈ C when some of its coefficients have been erased. We do that using the p -adic expansion of w and particular representations of the parity-check polynomial matrix of the code. From these matrix polynomial representations we recursively select certain equations that w must satisfy and have only coefficients in the field p r − 1 Z p r . We exploit the natural block Toeplitz structure of the sliding parity-check matrix to derive a step by step methodology to obtain a list of possible codewords for a given corrupted codeword w , that is, a list with the closest codewords to w. [ABSTRACT FROM AUTHOR]