In this paper, several infinite families of codes over the extension of non-unital non-commutative rings are constructed utilizing general simplicial complexes. Thanks to the special structure of the defining sets, the principal parameters of these codes are characterized. Specially, when the employed simplicial complexes are generated by a single maximal element, we determine their Lee weight distributions completely. Furthermore, by considering the Gray image codes and the corresponding subfield-like codes, numerous of linear codes over Fqare also obtained, whereqis a prime power. Certain conditions are given to ensure the above linear codes are (Hermitian) self-orthogonal in the case ofq= 2; 3; 4. It is noteworthy that most of the derived codes over Fqsatisfy the Ashikhmin-Barg’s condition for minimality. Besides, we obtain two infinite families of distanceoptimal codes over Fqwith respect to the Griesmer bound. By puncturing the Gray image codes and subfield-like codes, several classes of projective codes are presented.

In this paper, we construct infinitely many families of distance-optimal binary BCH codes with the minimum distance 6 and an infinite family of distance-optimal quaternary BCH codes with the minimum distance 4. We also construct several infinite families of cyclic and negacyclic BCH codes over${\mathbf { F}}_{2}$,${\mathbf { F}}_{3}$,${\mathbf { F}}_{4}$,${\mathbf { F}}_{5}$,${\mathbf { F}}_{7}$and${\mathbf { F}}_{9}$with good parameters$n,\,k,\,d$, such that the maximal possible minimum distance$d_{\max }$of a linear$[n, k]_{q}$code is at most$d_{\max } \leq d+8$. Many codes in these families have optimal or best known minimum distances. 145 optimal or best known codes are constructed as cyclic codes, negacyclic codes, their shortening codes and punctured codes. Several infinite families of rate$\frac {1}{2}$negacyclic$\left [{{n, \frac {n+1}{2}, d}}\right]_{q}$codes or$\left [{{n, \frac {n}{2}, d}}\right]_{q}$codes, such that their minimum distances satisfy$d\geq \frac {cn}{\log _{q} n}$, where c is a positive constant, are also constructed. These are first several families of such negacyclic codes reported in the literature.

In this paper, we investigate the power function$F(x)=x^{d}$over the finite field$\mathbb {F}_{2^{4n}}$, where$n$is a positive integer and$d=2^{3n}+2^{2n}+2^{n}-1$. We prove that this power function is AP$c\text{N}$with respect to all$c\in \mathbb {F}_{2^{4n}}\setminus \{1\}$satisfying$c^{2^{2n}+1}=1$, and we determine its$c$-differential spectrum. To the best of our knowledge, this is the second class of AP$c\text{N}$power functions over finite fields of even characteristic. By the same proof ideas, we completely determine the differential spectrum of this function, and give an affirmative answer to a recent conjecture proposed by Budaghyan, Calderini, Carlet, Davidova and Kaleyski.

In this paper, a class of p-ary 3-weight linear codes and a class of binary 2-weight linear codes are proposed respectively by virtue of the properties of the perfect nonlinear functions over Fp(m)and (m, s)-bent functions from F2(m)to F2(s), where p is an odd prime and m, s are positive integers. The weight distributions are completely determined by the sign of the Walsh transform of weakly regular bent functions and the size of the preimage of the employed (m, s)-bent functions at the zero point, respectively. As a special case, a class of optimal linear codes meeting Griesmer bound is obtained from our construction.