On a question of Babadi and Tarokh

TL;DR

This paper extends Babadi and Tarokh's spectral distribution results for binary linear block codes, showing that Gold sequences with smaller dual distances also exhibit spectral randomness, thus relaxing previous constraints.

Contribution

It improves the existing theorem on spectral randomness of binary sequences, specifically demonstrating Gold sequences satisfy the property with smaller dual distances.

Findings

Gold sequences with dual distance 5 satisfy spectral randomness

Theorem on spectral distribution is generalized to smaller dual distances

Numerical experiments support the relaxed conditions

Abstract

In a recent remarkable paper, Babadi and Tarokh proved the "randomness" of sequences arising from binary linear block codes in the sense of spectral distribution, provided that their dual distances are sufficiently large. However, numerical experiments conducted by the authors revealed that Gold sequences which have dual distance 5 also satisfy such randomness property. Hence the interesting question was raised as to whether or not the stringent requirement of large dual distances can be relaxed in the theorem in order to explain the randomness of Gold sequences. This paper improves their result on several fronts and provides an affirmative answer to this question.