Binary${m}$-sequences are those with the largest period$n=2^{m}-1$among the binary sequences produced by linear shift registers with length$m$. They have a wide range of applications in communication since they have several desirable pseudorandom properties, such as balance, uniform pattern distribution, and ideal (classical) autocorrelation. In 1967, Mandelbaum introduced a 2-adic version of classical autocorrelation of binary sequences, called arithmetic autocorrelation, in his research on arithmetic codes. Later, Goresky and Klapper generalized this notion to the nonbinary case and got several properties of arithmetic autocorrelation related to linear shift registers with carry. Recently, Z. Chen et al. showed an upper bound on the arithmetic autocorrelation of binary${m}$-sequences and raised a conjecture on the absolute value distribution of the arithmetic autocorrelation of binary${m}$-sequences. In this paper, we present a general formula for computing arithmetic autocorrelation, from which we completely determine the arithmetic autocorrelation distribution of arbitrary binary${m}$-sequences. In particular, the conjecture raised by Z. Chen et al. is verified.

Based on theoretical neuroscience, G. Cotardo and A. Ravagnani (2022) introduced a class of asymmetric binary codes called combinatorial neural codes (CN codes for short), with a “matched metric”$\delta _{r}$called asymmetric discrepancy, instead of the Hamming distance$d_{H}$for usual error-correcting codes. They also presented the Hamming, Singleton and Plotkin bounds for CN codes with respect to$\delta _{r}$and asked how to construct CN codes${\mathcal C}$with large size$| {\mathcal C}|$and minimum$\delta _{r}({\mathcal C})$. In this paper, we first show that a binary code${\mathcal C}$reaches one of the above bounds for$\delta _{r}({\mathcal C})$if and only if${\mathcal C}$reaches the corresponding bounds for$d_{H}$and$r$is sufficiently close to 1. This means that all optimal CN codes come from the usual optimal codes. Then, we present several constructions of CN codes with good and flexible parameters$(n,K, \delta _{r}({\mathcal C}))$by using bent functions.

The purpose of this paper is to determine the 4-adic complexity of the balanced quaternary sequences of period 2(2n−1) with ideal autocorrelation defined by Jang et al. (ISIT, pp. 278-281, 2009). Results show that the 4-adic complexity of such sequences is large enough to resist the attack of the rational approximation algorithm for feedback with carry shift registers.