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.

A Locally Recoverable Code is a code such that the value of any single coordinate of a codeword can be recovered from the values of a small subset of other coordinates. When we have δ non-overlapping subsets of cardinality ri that can be used to recover the missing coordinate we say that a linear code C with length n, dimension k, minimum distance d has (r1, . . . , rδ)locality and denote by [n, k, d; r1, r2, . . . , rδ]. In this paper we provide a new upper bound for the minimum distance of these codes. Working with a finite number of subgroups of cardinality ri+ 1 of the automorphism group of a function field F|Fqof genus g ≥ 1 we propose a construction of [n, k, d; r1, r2, . . . , rδ]-codes and apply the results to some well known families of function fields.

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.

For applications in algebraic geometric codes, an explicit description of bases of Riemann–Roch spaces of divisors on function fields over finite fields is needed. We give an algorithm to compute such bases for one-point divisors, and Weierstrass semigroups over an optimal tower of function fields. We also explicitly compute Weierstrass semigroups till level eight.