In this paper, we propose and studyr-minimal codes, a natural extension of minimal codes which have been extensively studied with respect to Hamming metric, rank metric and sum-rank metric. We first proposer-minimal codes in a general setting where the ambient space is a finite dimensional left module over a division ring and is supported on a lattice. We characterize minimal subcodes andr-minimal codes, derive a general singleton bound, and give existence results forr-minimal codes by using combinatorial arguments. We then considerr-minimal rank metric codes over a field extension E/F of degreem, where E can be infinite unless otherwise specified. We characterize these codes in terms of cuttingr-blocking sets, generalized rank weights of the codes and those of the dual codes, and classify codes whoser-dimensional subcodes have constant rank support weight. Next, with the help of the evasiveness property of cuttingr-blocking sets and some upper bounds for the dimensions of evasive subspaces, we derive several lower and upper bounds for the minimal length ofr-minimal codes. Furthermore, when E is finite, we establish a general upper bound which generalizes and improves the counterpart for minimal codes in the literature. As a corollary, we show that ifm= 3, then for anyk⩾ 2, the minimal length ofk-dimensional minimal codes is equal to 2k. To the best of our knowledge, whenm⩾ 3, there is no known explicit formula for the minimal length ofk-dimensional minimal codes for arbitrarykin the literature.

Let$\mathbf {H}$be the Cartesian product of a family of left modules over a ring$S$, indexed by a finite set$\Omega $. We study the MacWilliams extension property (MEP) with respect to$(\mathbf {P},\omega)$-weight on$\mathbf {H}$, where$\mathbf {P}=(\Omega,\preccurlyeq _{\mathbf {P}})$is a poset and$\omega:\Omega \longrightarrow \mathbb {R}^{+}$is a weight function. We first give a characterization of the group of$(\mathbf {P},\omega)$-weight isometries of$\mathbf {H}$, which is then used to show that MEP implies the unique decomposition property (UDP) of$(\mathbf {P},\omega)$, which, for the case that$\omega $is identically 1, further implies that$\mathbf {P}$is hierarchical. When$\mathbf {P}$is hierarchical or$\omega $is identically 1, with some weak additional assumptions, we give necessary and sufficient conditions for$\mathbf {H}$to satisfy MEP with respect to$(\mathbf {P},\omega)$-weight in terms of MEP with respect to Hamming weight. With the help of these results, when$S$is a finite field, we compare MEP with various well studied coding-theoretic properties including the property of admitting MacWilliams identity (PAMI), reflexivity of partitions, UDP, transitivity of the group of isometries and whether$(\mathbf {P},\omega)$induces an association scheme; in particular, we show that MEP is always stronger than all the other properties.

Let$\mathbf {H}$be the Cartesian product of a family of finite abelian groups. Via a polynomial approach, we give sufficient conditions for a partition of$\mathbf {H}$induced by weighted poset metric to be reflexive, which also become necessary for some special scenarios. Moreover, by examining the roots of the Krawtchouk polynomials, we give sufficient conditions for a partition of$\mathbf {H}$induced by combinatorial metric to be non-reflexive, and then give several examples of non-reflexive partitions. When$\mathbf {H}$is a vector space over a finite field$\mathbb {F}$, we consider the property of admitting MacWilliams identity (PAMI) and the MacWilliams extension property (MEP) for partitions of$\mathbf {H}$. More specifically, under some invariance assumptions, we show that two partitions of$\mathbf {H}$admit MacWilliams identity if and only if they are mutually dual and reflexive, and any partition of$\mathbf {H}$satisfying MEP is in fact an orbit partition induced by some subgroup of$\mathrm {Aut}\,_{\mathbb {F}}(\mathbf {H})$, which is necessarily reflexive. Furthermore, we show that the aforementioned non-reflexive partitions induced by combinatorial metric do not satisfy MEP, which further enables us to disprove a conjecture proposed by Pinheiro et al., (2019).

For $R \triangleq Ma{t_m}({\mathbb{F}})$, the ring of all m × m matrices over the finite field ${\mathbb{F}}$ with $|{\mathbb{F}}| = q$, and the left R-module $A \triangleq Ma{t_{m,k}}({\mathbb{F}})$ with m + 1 ⩽ k, by deriving the minimal length of solutions of the related isometry equation, Dyshko has proved in [3], [4] that the minimal code length n for Annot satisfying the MacWilliams extension property (MEP) with respect to Hamming weight is equal to $\prod\nolimits_{i = 1}^m {\left( {{q^i} + 1} \right)}$. In this paper, using the Möbius functions, we derive the minimal length of nontrivial solutions of the isometry equation for a finite lattice. For the finite vector space ${\mathbf{H}} \triangleq \prod\nolimits_{i \in \Omega } {{{\mathbb{F}}^{{k_i}}}}$, a poset P = (Ω, ≼P) and a map ω: Ω → ℝ+give rise to the (P, ω)-weight on H, which has been proposed by Hyun, Kim and Park in [18]. For such a weight, we study the relations between the MEP and other properties including admitting MacWilliams identity, Fourier-reflexivity of involved partitions and the Unique Decomposition Property (UDP) defined for (P, ω). We give necessary and sufficient conditions for H to satisfy the MEP with the additional assumption that either P is hierarchical or ω is identically 1, i.e., (P, ω)-weight coincides with P-weight, which further allow us to partly answer a conjecture proposed by Machado and Firer in [22].

Let H be the cartesian product of a family of abelian groups indexed by a nonempty finite set Ω. A given poset P = (Ω, ≼P) and a map ω : Ω → ℝ+give rise to the (P, ω)-weight on H, which further leads to a partition $\mathcal{Q}\left( {{\text{H}},{\text{P}},\omega } \right)$ of H. For the case that H is finite, we give sufficient conditions for two codewords to belong to the same block of Λ, the dual partition of $\mathcal{Q}\left( {{\text{H}},{\text{P}},\omega } \right)$, and sufficient conditions for $\mathcal{Q}\left( {{\text{H}},{\text{P}},\omega } \right)$ to be Fourier-reflexive. By relating the involved partitions with certain polynomials, we show that such sufficient conditions are also necessary if P is hierarchical and ω is integer valued. With H further set to be a finite vector space over a finite field $\mathbb{F}$, from a partition perspective, we extend the property of "admitting MacWilliams identity" to arbitrary pairs of partitions of H, and prove that a pair of $\mathbb{F}$-invariant partitions (Λ, Γ) with |Λ| = |Γ| admits MacWilliams identity if and only if (Λ, Γ) is a pair of mutually dual Fourier-reflexive partitions. Such a result is applied to the partition $\mathcal{Q}\left( {{\text{H}},{\text{P}},\omega } \right)$. Finally, with H set to be a (possibly infinite) left module over a ring S, we show that each (P, ω)- weight isometry of H uniquely induces an order automorphism of P, which further leads to a group homomorphism from the group of (P, ω)-weight isometries to Aut (P), whose kernel consists of isometries preserving the P-support.

In 1991, Wei proved a duality theorem that established an interesting connection between the generalized Hamming weights of a linear code and those of its dual code. Wei’s duality theorem has since been extensively studied from different perspectives and extended to other settings. In this paper, we re-examine Wei’s duality theorem and its various extensions, henceforth referred to as Wei-type duality theorems, from a new Galois connection perspective. Our approach is based on the observation that the generalized Hamming weights and the dimension/length profiles of a linear code form a Galois connection. The central result of this paper is a general Wei-type duality theorem for two Galois connections between finite subsets of$\mathbb {Z}$, from which all the known Wei-type duality theorems can be recovered. As corollaries of our central result, we prove new Wei-type duality theorems for$w$-demi-matroids defined over finite sets and$w$-demi-polymatroids defined over modules with a composition series, which further allows us to unify and generalize all the known Wei-type duality theorems established for codes endowed with various metrics.

Let$\mathbf {H}$be the cartesian product of a family of finite abelian groups indexed by a finite set$\Omega $. A given poset (i.e., partially ordered set)$\mathbf {P}=(\Omega,\preccurlyeq _{\mathbf {P}})$gives rise to a poset metric on$\mathbf {H}$, which further leads to a partition$\mathcal {Q}(\mathbf {H},\mathbf {P})$of$\mathbf {H}$. We prove that if$\mathcal {Q}(\mathbf {H},\mathbf {P})$is Fourier-reflexive, then its dual partition$\Lambda $coincides with the partition of$\hat {\mathbf {H}}$induced by$\mathbf {\overline {P}}$, the dual poset of$\mathbf {P}$, and moreover,$\mathbf {P}$is necessarily hierarchical. This result establishes a conjecture proposed by Gluesing-Luerssen in Gluesing-Luerssen, 2015. We also show that with some other assumptions,$\Lambda $is finer than the partition of$\hat {\mathbf {H}}$induced by$\mathbf {\overline {P}}$. In addition, we give some necessary and sufficient conditions for$\mathbf {P}$to be hierarchical, and for the case that$\mathbf {P}$is hierarchical, we give an explicit criterion for determining whether two codewords in$\hat {\mathbf {H}}$belong to the same block of$\Lambda $. We prove these results by relating the involved partitions with certain family of polynomials, a generalized version of which is also proposed and studied to generalize the aforementioned results.

Let$\mathrm{H}=\prod\nolimits_{i\in\Omega}H_{i}$be the cartesian product of finite abelian groups$H_{i}$indexed by a finite set$\Omega$. Any partition of H gives rise to a dual partition of its character group$\hat{\mathrm{H}}$. A given poset (i.e., partially ordered set) P on$\Omega$gives rise to the corresponding poset metric on H, which further leads to a partition$\Gamma$of H. We prove that if$\Gamma$is Fourier-reflexive, then its dual partition$\hat{\Gamma}$coincides with the partition of$\hat{\mathrm{H}}$induced by$\overline{\mathrm{P}}$, the dual poset of P, and moreover, P is necessarily hierarchical. This result establishes a conjecture proposed by Heide Gluesing-Luerssen in [4]. We also show that with some other assumptions,$\hat{\Gamma}$is finer than the partition of$\hat{\mathrm{H}}$induced by$\overline{\mathrm{P}}$. We prove these results by relating the partitions with certain family of polynomials, whose basic properties are studied in a slightly more general setting.