Trellis codes with a good distance profile constructed from expander graphs
Yubin Zhu, Zitan Chen
TL;DR
The paper derives Singleton-type bounds for trellis codes, showing that at a fixed time instant the column-distance can surpass the corresponding bound for convolutional codes. It then constructs trellis codes over constant-size alphabets by coupling a convolutional code with block codes on a sequence of bipartite Ramanujan expanders and mapping to a fixed field extension, achieving a rate-distance performance close to that of maximum-distance-profile convolutional codes. The approach yields constant-alphabet trellis codes with $d_{\mathrm{free}}$ and $d_j^c$ scaling near the Singleton limits for large block lengths, while avoiding exponential field-size growth. This combines strong distance properties with potential for practical encoding/decoding due to expander-based structure. The work leaves open questions on efficient algorithms to realize these codes in hardware and on achieving nearly-MDP performance with constant-size alphabets.
Abstract
We derive Singleton-type bounds on the free distance and column distances of trellis codes. Our results show that, at a given time instant, the maximum attainable column distance of trellis codes can exceed that of convolutional codes. Moreover, using expander graphs, we construct trellis codes over constant-size alphabets that achieve a rate-distance trade-off arbitrarily close to that of convolutional codes with a maximum distance profile. By comparison, all known constructions of convolutional codes with a maximum distance profile require working over alphabets whose size grows at least exponentially with the number of output symbols per time instant.