CRC-Aided Short Convolutional Codes and RCU Bounds for Orthogonal Signaling

We extend earlier work on the design of convolutional code-specific CRC codes to
$Q$
-ary alphabets, with an eye toward
$Q$
-ary orthogonal signaling. Starting with distance-spectrum optimal, zero-terminated,
$Q$
-ary convolutional codes, we design
$Q$
-ary CRC codes so that the CRC/convolutional concatenation is distance-spectrum optimal. The
$Q$
-ary code symbols are mapped to a
$Q$
-ary orthogonal signal set and sent over an AWGN channel with noncoherent reception. We focus on
$Q=4$
, rate-1/2 convolutional codes in our designs. The random coding union bound and normal approximation are used in earlier works as benchmarks for performance for distance-spectrum-optimal convolutional codes. We derive a saddlepoint approximation of the random coding union bound for the coded noncoherent signaling channel, as well as a normal approximation for this channel, and compare the performance of our codes to these limits. Our best design is within 0.6 dB of the RCU bound at a frame error rate of 10
⁻⁴
.