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.

In this paper, we provide two complementary results for cyclic codes of composite length. First, we give a general construction of cyclic codes of length nr from cyclic codes of length n and a lower bound on the minimum distance. Numerical data show that many cyclic codes of composite length with the best parameters can be obtained in this way. Second, in the other direction, we show that for cyclic codes of length n with a primitive n-th root of unity as a non-zero, the minimum distance is roughly bounded by the square-free part of n. This means that we shall not expect good cyclic codes of length n in general if, for example, the length n is divisible by a high power of a prime.

MDS codes and self-dual codes are important families of classical codes in coding theory. Therefore, it is of interest to investigate MDS self-dual codes. The existence of MDS selfdual codes over finite field Fqis completely solved for q is even. In the literature, there are many known constructions of MDS self-dual codes for q is odd. In this paper, we present a unified approach on the existence of MDS self-dual codes with concise statements and simplified proof. It is illustrated that some of known results can also be stated in this framework. Furthermore, we can obtain some new MDS self-dual codes, especially for the case when q is not a square.

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, a family of p-ary cyclic codes whose duals have three pairwise nonconjugate zeros is proposed. The weight distribution of this family of cyclic codes is determined. It turns out that the proposed cyclic codes have five nonzero weights.

Cyclic codes are a subclass of linear codes and have wide applications in consumer electronics, data storage systems, and communication systems due to their efficient encoding and decoding algorithms. Cyclic codes with many zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a very small number of cases. In general, the calculation of the weight distribution of cyclic codes is heavily based on the evaluation of some exponential sums over finite fields. Very recently, Li studied a class of p-ary cyclic codes of length p2m-1, where p is a prime and m is odd. They determined the weight distribution of this class of cyclic codes by establishing a connection between the involved exponential sums with the spectrum of Hermitian forms graphs. In this paper, this class of p-ary cyclic codes is generalized and the weight distribution of the generalized cyclic codes is settled for both even m and odd m along with the idea of Li The weight distributions of two related families of cyclic codes are also determined.

In this paper, the value of Imax(C) for codebooks C = C(D) constructed by certain almost difference sets D in Fqxis determined and expressed in terms of Jacobi sums from which it shows that such codebooks nearly meet the Welch bound. This result is an answer of a question raised by C.Ding and T.Feng in [2]. We also present another series of codebooks which nearly meet the Welch bound, where the codebook is constructed by a subset R = Fq1⊕ Fq2. When q1= q2, Dεis an almost difference set of R.