Classical Goppa codes are a well-known class of codes with applications in code-based cryptography, which are a special case of alternant codes. Many papers are devoted to the search for Goppa codes with a cyclic extension or with a cyclic parity-check subcode. Let$\Bbb F_{q}$be a finite field with$q=2^{l}$elements, where l is a positive integer. In this paper, we determine all the generator polynomials of cyclic expurgated or extended Goppa codes under some prescribed permutations induced by the projective general linear automorphism$A \in PGL_{2}(\Bbb F_{q})$. Moreover, we provide some examples to support our findings.

Let Fqbe a finite field with q elements and p be a prime with gcd(p, q) = 1. Let G be a finite abelian p-group and Fq(G) be a group algebra. In this paper, we find all primitive idempotents and minimal abelian group codes in the group algebra Fq(G). Furthermore, we give all LCD abelian codes (linear code with complementary dual) and self-orthogonal abelian codes of Fq(G).

Let$\Bbb F_{p}$be a finite field with$p$elements, where$p$is a prime. Let$N \ge 2$be an integer and$d$be the least positive integer satisfying$p^{d} \equiv -1 \pmod N$. Let$q = p^{2sd}$for some integers$s$. In some special cases, we obtain the explicit evaluation of the following exponential sums:$S(a,b)=\sum _{x\in \Bbb F_{q}^{*}}\zeta _{p}^{ \mathrm {Tr}_{q/p}(ax^{(({q-1})/{N})}+bx)}$. As applications, Walsh spectrums of the monomial functions$\mathrm {Tr}_{q/p}(x^{(({q-1})/{N})})$in three cases are investigated. Our results show that Walsh spectrums of the monomial functions have at most four, five, or seven distinct values. Furthermore, three families of the monomial functions with three-valued Walsh spectrums are presented, seeCorollaries 12,21,31, and32. Consequently, certain previously known results by Li and Yue and Moisio are extended.

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. In this paper, let Fr be a finite field with r = qm. Suppose that g1, g2 ∈ F*rare not conjugates over Fq, ord(g1) = n1, ord(g2) = n2, d = gcd(n1, n2), and n = n1n2/d. Let Fq(g1) = Fqm1, Fq(g2) = Fqm2, and Ti denote the trace function from Fqmito Fq for i = 1, 2. We define a cyclic code C(q,m,n1,n2) = {c(a, b) : a ∈ Fqm1, b ∈ Fqm2}, where c(a, b) = (T1(ag01) + T2(bg02), T1(ag11) + T2(bg12), ... , T1(agn-11) + T2(bgn-12)). We mainly use Gauss periods to present the weight distribution of the cyclic code C(q,m,n1,n2). As applications, we determine the weight distribution of cyclic code C(q,m,qm1-1,qm2-1) with gcd(m1, m2) = 1; in particular, it is a three-weight cyclic code if gcd(q -1, m1 -m2) = 1. We also explicitly determine the weight distributions of some classes of cyclic codes including several classes of four-weight cyclic codes.