In this research paper, we present a novel method for obtaining the patterns of inputs that generate low-weight codewords in Recursive Systematic Convolutional (RSC) codes, hereafter referred to as low-weight inputs. RSC codes are commonly used as component codes for various concatenated schemes, including the popular Turbo codes. Efficient interleaver design is crucial in improving the error-correcting performance of Turbo codes; therefore, understanding the low-weight input patterns specific to the selected RSC code is essential. Existing graph-based search algorithms face increasing complexity as the constraint length of the RSC code increases. To address this issue, we introduce a method that directly derives the critical low-weight input patterns (up to Hamming weight 4) from the generator function of the RSC code. The proposed method addresses the complexity issue and can be extended to non-recursive systematic convolutional (NRSC) codes by modifying the generator function. To validate our novel method, we compare the union bound obtained using our approach to that obtained via the transfer function method and the simulation results for selected RSC codes. This comparison demonstrates the effectiveness of our method, which shows no significant counting losses.

In this paper, we present a novel design framework for Turbo codes (TCs). Referring to the generator functions of the employed recursive systematic convolutional (RSC) codes, we develop a method to determine not only the multiplicity, but also detailed patterns of low-weight (LW) input and LW parity-check (LWPC) sequences for the component codes. Next, we select the component RSC code based on the knowledge obtained by the proposed method and derive expressions necessary to evaluate the free distance dfof the resultant TC. Finally, we propose a novel interleaver class, named Coset interleaver, with its parameter selection criteria to enlarge df. Based on the proposed design framework, we succeed in designing a rate 1=3 TC with df= 42 for frame size N = 1024. At an Eb=N0of 2dB, our proposed design demonstrates a bit-error rate (BER) of 1:232 × 1010, which represents a significant improvement of 6:18 × 108and 2:207 108when compared to the BERs exhibited by the Quadratic Permutation Polynomial (QPP, df= 30) and S-random (df= 21) interleavers, respectively.