The symbol-pair coding theory was put forward by Cassuto and Blaum [IEEE TIT, 2011] to be applicable in high-density storage situations. Yaakobiet al. [IEEE TIT, 2016] extended the concept of symbol-pair metric tob-symbol metric whenb≥ 2. The extensive research on theb-symbol Hamming weight spectra of cyclic codes has been centered on the case where the alphabet is a finite field. The case of cyclic codes over Z4, an extremely important class of codes, has been overlooked for a long time in the exploration of theb-symbol Hamming weight spectra. In this paper, we study theb-symbol Hamming weight spectra of the shortened Kerdock codesK−mand the Kerdock codesKmover Z4. The formulas for calculating the symbol-pair Hamming weight of the codewords inK−mandKmare given, and their values hinge on the trace-like function values of specific elements in the Teichmüller set. In particular, we present a class of Z4-cyclic codes with three non-zerob-symbol Hamming weights. As by-products, theb-symbol Hamming weight hierarchies of the Preparata codes and the Goethals codes are provided.

A code$C$is called$Z_{p}Z_{p^{2}}$-linear if it is the Gray image of a$Z_{p}Z_{p^{2}}$-additive code. For any prime number$p$larger than 3, the bounds of the rank of$Z_{p}Z_{p^{2}}$-linear codes are given. For each value of the rank and the pairs of rank and the dimension of the kernel of$Z_{p}Z_{p^{2}}$-linear codes, we give detailed construction of the corresponding codes. As an example, the rank and the dimension of the kernel of$Z_{5}Z_{25}$-linear codes are studied.

The Galois ring GR$(4^{\Delta})$is the residue ring$Z_{4}[x]/(h(x))$, where$h(x)$is a basic primitive polynomial of degree$\Delta $over$Z_{4}$. For any odd$\Delta $larger than 1, we construct a partition of GR$(4^{\Delta}) \backslash \{0\}$into 6-subsets of type$\{a,b,-a-b,-a,-b,a+b\}$and 3-subsets of type$\{c,-c,2c\}$such that the partition is invariant under the multiplication by a nonzero element of the Teichmuller set in GR$(4^{\Delta})$and, if$\Delta $is not a multiple of 3, under the action of the automorphism group of GR$(4^{\Delta})$. As a corollary, this implies the existence of quasi-cyclic additive 1-perfect codes of index$(2^{\Delta} -1)$in$D((2^{\Delta} -1)(2^{\Delta} -2)/{6}, 2^{\Delta} -1)$where$D(m,n)$is the Doob metric scheme on$Z^{2m+n}$.

Upper and lower bounds on the largest number of weights in a cyclic code of given length, dimension and alphabet are given. An application to irreducible cyclic codes is considered. Sharper upper bounds are given for the special cyclic codes (called here strongly cyclic), whose nonzero codewords have period equal to the length of the code. Asymptotics are derived on the function Γ(k, q), that is defined as the largest number of nonzero weights a cyclic code of dimension k over Fq can have, and an algorithm to compute it is sketched. The nonzero weights in some infinite families of Reed-Muller codes, either binary or q-ary, as well as in the q-ary Hamming code are determined, two difficult results of independent interest.