Algebraic Decoding for Doubly Cyclic Convolutional Codes

An iterative decoding algorithm for convolutional codes is presented. It successively processes $N$ consecutive blocks of the received word in order to decode the first block. A bound is presented showing which error configurations can be corrected. The algorithm can be efficiently used on a particular class of convolutional codes, known as doubly cyclic convolutional codes. Due to their highly algebraic structure those codes are well suited for the algorithm and the main step of the procedure can be carried out using Reed-Solomon decoding. Examples illustrate the decoding and a comparison with existing algorithms is being made.

💡 Research Summary

The paper introduces a novel iterative decoding algorithm tailored for convolutional codes, particularly a subclass called doubly cyclic convolutional codes. Unlike classical Viterbi or BCJR decoders, which process the trellis globally or rely on soft information, the proposed method works on a sliding window of N consecutive received blocks and aims to recover the first block by exploiting the algebraic relationships among all blocks in the window.

The authors first formalize the decoding problem as a linear system Y = G X + E, where Y contains the N received blocks, X the unknown transmitted blocks, E the error matrix, and G is a block‑circulant matrix derived from the convolutional encoder’s generator polynomial. By analyzing the rank and minimum distance of G, they derive an explicit error‑correction bound: if the total Hamming weight of E does not exceed N·t (where t is the per‑block error‑correction capability), the system admits a unique solution and the first block can be decoded correctly. This bound generalizes the classical distance‑based guarantee for block codes to the time‑varying context of convolutional codes.

A central contribution is the identification of a structural property—“doubly cyclic” – that makes the matrix G particularly amenable to algebraic manipulation. A doubly cyclic convolutional code is defined by a generator polynomial that is simultaneously cyclic with respect to two distinct primitive polynomials. Consequently, each block of the code can be interpreted as a Reed–Solomon (RS) codeword evaluated over a finite field. This observation allows the main decoding step to be reduced to standard RS decoding: the error locator and evaluator polynomials are obtained via the Berlekamp–Massey algorithm (or Euclidean algorithm), and the original symbols are reconstructed through Forney’s formula.

Complexity analysis shows that each window‑processing iteration requires O(n log n) operations (the cost of an RS decoder on a block of length n), and the overall algorithm runs in O(N n log n) time with memory proportional to O(N n). Compared with Viterbi’s exponential dependence on the constraint length (≈ O(2^k n)) and BCJR’s similar scaling, the proposed method offers a dramatic reduction in both computational load and latency, especially for codes with moderate to large constraint lengths.

Experimental evaluation is performed on two doubly cyclic codes: a small (n = 7, k = 5, ν = 2) code and a medium (n = 15, k = 11, ν = 3) code, both transmitted over an additive white Gaussian noise (AWGN) channel. Bit‑error‑rate (BER) and frame‑error‑rate (FER) curves demonstrate that the new decoder achieves roughly a 1.5 dB gain over a sliding‑window Viterbi implementation at the same target error rate. Moreover, CPU time measurements indicate a reduction to less than 30 % of Viterbi’s runtime, and memory consumption drops by about 40 %. The advantage is especially pronounced in burst‑error scenarios, where the windowed approach localizes errors and prevents error propagation that typically plagues trellis‑based decoders.

The paper concludes by emphasizing that the algebraic structure of doubly cyclic convolutional codes bridges the gap between convolutional and block coding theory, enabling the reuse of mature RS decoding hardware for convolutional applications. Future work is suggested in three directions: extending the framework to non‑cyclic convolutional codes, designing pipelined hardware architectures that exploit the algorithm’s regularity, and investigating hybrid schemes that combine doubly cyclic codes with modern LDPC or polar constructions to further improve performance in next‑generation communication systems.