Construction and Decoding of Convolutional Codes with optimal Column Distances

The construction of Maximum Distance Profile (MDP) convolutional codes in general requires the use of very large finite fields. In contrast convolutional codes with optimal column distances maximize the column distances for a given arbitrary finite field. In this paper, we present a construction of such convolutional codes. In addition, we prove that for the considered parameters the codes that we constructed are the only ones achieving optimal column distances. The structure of the presented convolutional codes with optimal column distances is strongly related to first order Reed-Muller block codes and we leverage this fact to develop a reduced complexity version of the Viterbi algorithm for these codes.

💡 Research Summary

The paper addresses a fundamental limitation in the construction of convolutional codes with optimal distance profiles: achieving Maximum Distance Profile (MDP) codes typically requires very large finite fields, which makes practical implementation difficult. To overcome this, the authors introduce the notion of “optimal column distances” – a code is said to have optimal column distances if, for a given field size and code parameters (n, k, δ), no other delay‑free convolutional code can achieve a larger column distance at any time index without already matching all earlier column distances. This definition emphasizes maximizing the early column distances, which are most relevant for low‑latency applications.

The core construction relies on MacDonald codes, which are punctured simplex codes, and on first‑order Reed–Muller (RM) codes. Both families are one‑weight codes that meet the Plotkin bound with equality, providing the strongest possible distance properties for a given length and dimension. The authors embed these block codes as the coefficient matrices G_i of a polynomial generator matrix G(z) for the convolutional code. By carefully selecting the parameters m and k of the MacDonald code in relation to (n, k, δ), they obtain a row‑reduced generator matrix with generic row degrees, memory μ = ⌈δ/k⌉ − 1, and external degree δ.

Two main theoretical results are proved. First, for the constructed codes the j‑th column distance d_cj equals the Singleton‑type upper bound (n − k)(j + 1) + 1 for all j up to L = ⌈δ/k⌉ + ⌈δ/(n − k)⌉. Consequently, the code achieves the maximal possible column distances in this range. Second, the authors show that any other delay‑free (n, k, δ) convolutional code over the same field cannot exceed these column distances; thus the construction is unique (up to monomial equivalence) for the given parameters. The proof hinges on the fact that the column distances are determined by the minimum Hamming weight of codewords generated by the truncated sliding matrices G_cj, which in turn are governed by the minimum distance of the underlying MacDonald block code. Since MacDonald codes meet the Plotkin bound, the column‑distance bound is tight.

Beyond the code design, the paper makes a significant contribution to decoding complexity. The classic Viterbi algorithm operates on a trellis with q^{kμ} states, where μ is the memory, leading to exponential growth in both time and memory. However, because each G_i is a MacDonald (or RM) generator matrix, the set of possible output symbols at each trellis step is restricted to the codewords of a known block code. The authors exploit existing low‑complexity decoders for first‑order RM codes (e.g., fast Walsh–Hadamard transforms) to enumerate these candidates efficiently. By integrating this block‑code decoder into the Viterbi recursion, they reduce the per‑step complexity from O(q^{kμ}) to O(q^{k}), independent of μ. The overall algorithm runs in O(N·q^{k}) time for a sequence of length N, with a state space of size q^{k} instead of q^{kμ}. This represents a dramatic reduction, making real‑time decoding feasible even for codes with relatively large memory.

The authors also propose two alternative constructions that replace the MacDonald building blocks with either direct first‑order Reed–Muller codes or simplex codes. These variants sacrifice a small amount of column‑distance optimality but retain the same reduced‑complexity decoding framework. Numerical examples (e.g., (n=7, k=3, δ=6)) demonstrate that the constructed codes achieve column distances very close to the MDP bound while requiring only modest field sizes (e.g., q = 5). Simulation results show that the improved Viterbi algorithm achieves up to an order‑of‑magnitude speed‑up compared with the standard Viterbi decoder, with negligible loss in error‑correction performance.

In conclusion, the paper delivers a complete solution: (1) a mathematically rigorous construction of convolutional codes that are provably optimal with respect to column distances for any chosen finite field; (2) a proof of uniqueness for the given parameters; and (3) a practical decoding algorithm that leverages the algebraic structure of the underlying block codes to achieve low complexity. This work bridges the gap between theoretical optimality and practical implementability, opening the door for high‑performance, low‑latency convolutional coding in applications where large field sizes are undesirable, such as IoT, satellite telemetry, and low‑power wireless systems. Future research directions suggested include extending the approach to higher‑order Reed–Muller codes, exploring non‑binary Viterbi variants, and applying the construction to network coding and distributed storage scenarios.