In recent years, linear complementary pairs (LCPs) of codes and linear complementary dual (LCD) codes have gained significant attention due to their applications in coding theory and cryptography. In this work, we construct explicit LCPs of codes and LCD codes from function fields of genus$g \geq 1$. To accomplish this, we present pairs of suitable divisors that give rise to non-special divisors of degree$g-1$in the function field. The results are applied in constructing LCPs of algebraic geometry codes and LCD algebraic geometry (AG) codes in Kummer extensions, hyperelliptic function fields, and elliptic curves.

A flag of codes$C_{0} \subsetneq C_{1} \subsetneq \cdots \subsetneq C_{s} \subseteq \mathbb {F}_{q} ^{n}$is said to satisfy theisometry-dual propertyif there exists${\mathbf{x}}\in (\mathbb {F}_{q}^{*})^{n}$such that the code$C_{i}$isx-isometric to the dual code$C_{s-i}^\perp $for all$i=0,\ldots, s$. For$P$and$Q$rational places in a function field$\mathcal {F}$, we investigate the existence of isometry-dual flags of codes in the families of two-point algebraic geometry codes$C_{\mathcal {L}}(D, a_{0}P+bQ)\subsetneq C_{\mathcal {L}}(D, a_{1}P+bQ)\subsetneq {\dots } \subsetneq C_{\mathcal {L}}(D, a_{s}P+bQ)$, where the divisor$D$is the sum of pairwise different rational places of$\mathcal {F}$and$P, Q$are not in$\mathop {\mathrm {supp}}\nolimits (D)$. We characterize those sequences in terms of$b$for general function fields. We then apply the result to the broad class of Kummer extensions$\mathcal {F}$defined by affine equations of the form$y^{m}=f(x)$, for$f(x)$a separable polynomial of degree$r$, where$\gcd (r, m)=1$. For$P$the rational place at infinity and$Q$the rational place associated to one of the roots of$f(x)$, and for$D$an$Aut(\mathcal {F}/ \mathbb {F}_{q})$-invariant sum of rational places of$\mathcal {F}$, such that$P, Q \notin \mathop {\mathrm {supp}}\nolimits D$, it is shown that the flag of two-point algebraic geometry codes has the isometry-dual property if and only if$m$divides$2b+1$. At the end we illustrate our results by applying them to two-point codes over several well know function fields.

In this paper, we construct some class of explicit subcovers of the curve X n,r defined over F q(n) by affine equation y q(n-1) + ··· + y q + y = x q(n-r)+1 - x q(n)+q(n-r) . These subcovers are defined over F q(n) by affine equation g s (y) = x q(n)+q(n-r) -x q(n-r) +1, where g s (y) is a q-polynomial of degree q s . The Weierstrass semigroup H(P ∞ ), where P ∞ is the only point at infinity on such subcovers, is determined for 1 ≤ s ≤ 2r - n + 1, and the corresponding one-point AG codes are investigated. Codes establishing new records on the parameters with respect to the previously known ones are discovered, and 108 improvements on MinT tables are obtained.

We determine the Weierstrass semigroup of a pair of certain rational points on the Giulietti-Korchmáros maximal curves. We use this semigroup to obtain two-point algebraic geometric (AG) codes with better parameters than comparable one-point AG codes arising from these curves. These parameters are new records in the MinT's tables.

We compute the Weierstrass semigroup at one totally ramified place for Kummer extensions defined by ym= f (x)λ, where f (x) is a separable polynomial over Fq. In addition, we compute the Weierstrass semigroup at two certain totally ramified places. We then apply our results to construct one- and two-point algebraic geometric codes with good parameters.

We determine the full automorphism group of the Generalized Hermitian curve, denoted by GH, which generalizes the Hermitian curve. The automorphism group of a code is important in Coding Theory, and in this way, we determine completely the automorphism group of the one-point AG codes over the GH curve.