Constantinescu et al. introduced the homogeneous weight on the integer residue ring$\mathbb {Z}_{m}$which can reflect more information compared with the Hamming weight. Few homogeneous weight linear codes over finite chain rings have important applications in cryptography, lattices, modular forms and combinatorics. In this paper, we construct an infinite class of cyclic codes over the finite chain ring$\mathbb {F}_{p^{t}}[\omega]/(\omega ^{2})$by the trace function, and determine their homogeneous weight distributions by applying the theory of exponential sums. In order to investigate the minimality of linear codes over finite chain rings, we firstly present the necessary and sufficient condition for linear codes over the finite chain ring$\mathbb {F}_{p^{t}}[\omega]/(\omega ^{2})$to be minimal or almost minimal by the Hamming weights of codewords. Then, based on the proposed condition and few Hamming weight cyclic codes, we give several classes of minimal and almost minimal linear codes. Furthermore, we derive several families of strongly regular graphs, strongly walk-regular graphs and triple sum sets by few homogeneous weight linear codes.

Let m be an arbitrary positive integer and D8mbe the dihedral group of order 8m, i.e., D8m= (x, y | x4m= 1, y2= 1, yxy = x-1). Left ideals of the dihedral group algebra F2[D8m] are called binary left dihedral codes of length 8m, and abbreviated as binary left D8m-codes. In this paper, we give an explicit representation and enumeration for all distinct self-dual binary left D8m-codes. These codes make up an important class of self-dual binary [8m, 4m]-codes such that the dihedral group D8mis necessarily a subgroup of the automorphism group of each code. In particular, we provide recursive algorithms to solve congruence equations over finite chain rings for constructing all distinct self-dual binary left D8m-codes and obtain a Mass formula to count the number of all these self-dual codes. As a preliminary application, we obtain the extremal self-dual binary [48, 24, 12]-code and an extremal self-dual binary [56, 28, 12]code from self-dual binary left D48-codes and left D56-codes respectively.

Let G(m,3,r)= (x, y | xm= 1, y3= 1, yx = xry) be a metacyclic group of order 3m, where gcd(m, r) = 1, 13≡ 1 (mod m). Then, left ideals of the group algebra Fq[G(m,3,r)] are called left metacyclic codes over Fq of length 3m, and abbreviated as left G(m,3,r)-codes. A system theory for left G(m,3,r)-codes is developed for the case of gcd(m, q) = 1 and r ≡ qE(mod m) for some positive integer ε, only using finite field theory and basic theory of cyclic codes and skew cyclic codes. The fact that any left G(m,3 1)-code is a direct sum of concatenated codes with inner codes λiand outer codes Ciis proved, where . Aiis a minimal cyclic code over Fqof length m and Ciis a skew cyclic code of length 3 over an extension field of Fq. Then, an explicit expression for each outer code in any concatenated code is provided. Moreover, the dual code of each left G(m,3,r)-code is given and self-orthogonal left G(m,3,r)-codes are determined.