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Bounds on List Decoding of Rank-Metric Codes
So far, there is no polynomial-time list decoding algorithm (beyond half the
minimum distance) for Gabidulin codes. These codes can be seen as the
rank-metric equivalent of Reed--Solomon codes. In this paper, we provide bounds
on the list size of rank-metric codes in order to understand whether
polynomial-time list decoding is possible or whether it works only with
exponential time complexity. Three bounds on the list size are proven. The
first one is a lower exponential bound for Gabidulin codes and shows that for
these codes no polynomial-time list decoding beyond the Johnson radius exists.
Second, an exponential upper bound is derived, which holds for any rank-metric
code of length and minimum rank distance . The third bound proves that
there exists a rank-metric code over \Fqm of length such that the
list size is exponential in the length for any radius greater than half the
minimum rank distance. This implies that there cannot exist a polynomial upper
bound depending only on and similar to the Johnson bound in Hamming
metric. All three rank-metric bounds reveal significant differences to bounds
for codes in Hamming metric.Comment: 10 pages, 2 figures, submitted to IEEE Transactions on Information
Theory, short version presented at ISIT 201
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