Distance Properties of Punctured Simplex Codes and Design of High-Rate PRC-LDPC Codes for Complexity-Constrained Applications

Cyclic simplex codes are usually deemed as impractical because of their extremely low rate and large codeword length for practical values of the code dimension. To address these limitations, researchers have resorted to punctured simplex codes, focusing on primitive polynomials and employing statistical analyses to investigate their properties. This paper delves deeper into the properties of punctured binary simplex codes, also focusing on the recently introduced family of Primitive Rate-Compatible Low-Density Parity-Check (PRC-LDPC) codes. We study the average behavior of punctured simplex codes in terms of minimum distance properties. Furthermore, our results highlight the potential of high-rate PRC-LDPC codes to reach or even surpass the performance of state-of-the-art code families. We show how to design good codes by applying puncturing and shortening operations to cyclic simplex codes, also considering complexity-constrained scenarios.