## Aspects of list-of-two decoding by Jonas Eriksson.

By Jonas Eriksson.

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Calculation of the complete coset weight distribution of a code is generally a hard task, so any information provided by the βi :s is of course valuable. The βi :s display some more or less straightforward properties worth noting. For any linear q-ary code of length n and error-correction capability t we have (τ ) βi = 0 for i ∈ {0, 1, . . , t} and (τ ) βi = n (q − 1)i for i ∈ {τ + 1, τ + 2, . . , n}. i The ﬁrst part is a consequence of the deﬁnition of the error-correction capability t. All error patterns of weight less than or equal to t are correctable.

Denote by rs the smallest number of such groups needed in order to obtain ls + 1. We then at least the correct number of unknowns. That is rs = k−1 have rs −1 s+1 (k − 1)(i + 1) > n 2 i=0 or equivalently s+1 k−1 rs (rs + 1) > n . 2 2 Since rs must be a positive integer we conclude rs = 1 + 2 2n s + 1 1 + 4 k−1 2 . ls . For the exact value of ls Thus we have determined the value of k−1 we continue by noting that the last (k − 1)-sized group of partitions is not necessarily completely used up. 3.

T + s, all yield a well behaved error locator polynomial. He also noted that each such error locator polynomial corresponds to an error pattern of weight t + s belonging to the coset indicated by the known syndromes. This fact—that we allow more than one solution—is typical of a list decoder. The result is in a way equivalent with a list decoding were the list decoding radius is precisely equal to the distance from the received polynomial and the closest codeword polynomial. In an actual list decoder, however, the list decoding radius is not dependent of the received polynomial.