Trellis codes with a good distance profile constructed from expander graphs

TL;DR

This paper establishes bounds on trellis code distances, demonstrates that trellis codes can outperform convolutional codes in column distance at a given time, and introduces expander graph-based constructions that achieve near-optimal rate-distance trade-offs over constant-size alphabets.

Contribution

It provides Singleton-type bounds for trellis codes, and presents a novel construction using expander graphs that attains near-optimal performance with fixed alphabet size.

Findings

Trellis codes can have higher column distances than convolutional codes at the same instant.

Expander graph-based trellis codes achieve near-optimal rate-distance trade-offs.

Constructed codes work over constant-size alphabets, unlike previous convolutional code constructions.

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.