Optimized Rate-Adaptive Error Correction through Puncturing and Shortening of Simplex Codes

This paper presents a strategy for obtaining efficient rate-adaptive error correction by puncturing and shortening simplex codes. We derive a family of punctured cyclic codes from a parent cyclic simplex code. Using a polynomial approach based on the parity-check polynomial \( h(x) \), we analyze key properties of these codes. We introduce a procedure to optimize code design with predefined values of the code rate \( R \) and length \( n \) through optimal multiple code shortening, aiming to increase the minimum distance by eliminating residual minimum distance codewords after puncturing operations. The effectiveness of this approach is demonstrated through practical examples, with BER vs. SNR curves obtained through numerical simulations.