Decoding error probability in the erasure channel and the $\boldsymbol~r$-th support weight distribution

Inspired by a recent work of Shen and Fu (2019), we study the decoding error probability of linear codes over the erasure channel under three decoding principles. We first simplify the explicit formulas obtained by Shen and Fu (2019), showing a much more direct relation to the $r$-th support weight distribution of linear codes. Then we compute the average $r$-th support weight distribution of $[n,k]_q$ linear codes. We resolve an open question of Shen and Fu (2019) concerning the error exponent of the average unsuccessful decoding probability under maximum likelihood decoding for random $[n,nR]_q$ linear codes. We also obtain the error exponent of the average unsuccessful decoding probability for random $[n,nR]_q$ linear codes under list decoding.