Group randomness properties of pseudo-noise and gold sequences

In this paper, we study the group randomness of pseudo-random sequences based on shortened first-order Reed-Muller codes and the Gold sequences. In particular, we characterize the empirical spectral distribution of random matrices from shortened first-order Reed-Muller codes. We show that although these sequences have very appealing randomness properties across individual codewords, they do not possess certain group randomness properties of i.i.d. sequences. In other words, the spectral distribution of random matrices from these sequences dramatically differs from that of the random i.i.d. generated matrices. In contrast, Gold sequences manifest the group randomness properties of random i.i.d. sequences. Upper bounds on the Kolmogorov complexity of these sequences are established, and it has been shown that these bounds are much lower than those of the random i.i.d. sequences, when the sequence length is large enough. We discuss the implications of these observations and motivate the need to develop novel randomness tests encompassing both individual and group randomness of sequences.