  Recently, rate-$1/\omega$ zero-terminated and tail-biting convolutional codes
(ZTCCs and TBCCs) with cyclic-redundancy-check (CRC)-aided list decoding have
been shown to closely approach the random-coding union (RCU) bound for short
blocklengths. This paper designs CRCs for rate-$(\omega-1)/\omega$ CCs with
short blocklengths, considering both the ZT and TB cases. The CRC design seeks
to optimize the frame error rate (FER) performance of the code resulting from
the concatenation of the CRC and the CC. Utilization of the dual trellis
proposed by Yamada \emph{et al.} lowers the complexity of CRC-aided serial list
Viterbi decoding (SLVD) of ZTCCs and TBCCs. CRC-aided SLVD of the TBCCs closely
approaches the RCU bound at a blocklength of $128$.
