Endcoding Complexity Versus Minimum Distance

Abstract

A bound on the minimum distance of a binary error-correcting code is established given constraints on the computational time-space complexity of its encoder where the encoder is modeled as a branching program. The bound obtained asserts that if the encoder uses linear time and sublinear memory in the most general sense, then the minimum distance of the code cannot grow linearly with the block length when the rate is nonvanishing, that is, the minimum relative distance of the code tends to zero in such a setting. The setting is general enough to include nonserially concatenated turbo-like codes and various generalizations. Our argument is based on branching program techniques introduced by Ajtai. The case of constant-depth AND-OR circuit encoders with unbounded fanins are also considered.