UCLA

This thesis consists of two main sections investigating the performance of cyclic-redundancy-check-aided (CRC-aided) list decoding on short block codes. The first section analyzes the performance of tail biting convolutional codes with CRC (CRC-TBCCs) and polar codes with CRC (CRC-Polar) with an eye toward the 5G standard. The second section concerns designing optimal CRC-convolutional codes for nonbinary orthogonal noncoherent signaling.

The first section focuses on designing a code for the physical broadcast channel of the 5G standard. The 5G standard encodes a 32-bit message with a 24-bit CRC and a (512, 32+24) polar code, with bit repetition to arrive at a final blocklength of 864 bits. We design shorter CRCs for this polar code in order to improve its performance. We also design low rate CRC-TBCCs with 32 bit messages as an alternative to the CRC-Polar in the 5G PBCH. CRCs are designed to optimize the distance spectrum of the concatenated CRC-Polar or CRC-TBCC. We call these CRCs distance-spectrum-optimal (DSO). We consider both adaptive and nonadaptive list decoders for these codes and compare their performance and complexity. Simulation results show that our CRC-TBCC and CRC-Polar designs significantly outperform the polar code in the 5G standard, with some CRC-TBCC designs closely approaching the random coding union (RCU) bound.

The second section presents designs for CRC-TBCCs and zero-terminated convolutional codes with CRC (CRC-ZTCCs) for communication with noncoherent orthogonal signaling. We design Q-ary convolutional codes to maximize the minimum distance, and then design Q-ary DSO CRCs for these convolutional codes, extending the work of Lou et. al. and Yang et. al. to nonbinary fields. The Q-ary code symbols are mapped to a Q-ary orthogonal signal set and sent over an AWGN channel with noncoherent reception. We consider cases where Q is a power of 2. We also derive a saddlepoint approximation for the calculation of the RCU bound for this channel. The RCU bound is a useful benchmark for the performance of CRC-convolutional codes, and we compare the performance of our codes to this bound.