Optimal Lee-Metric Anticodes

• Optimal Lee-metric anticodes are submodules over finite chain rings that achieve maximal size under a fixed Lee-weight constraint using block-diagonal generator matrices.
• They are characterized by support subtypes and weak compositions, which partition anticode families through a distributive lattice structure.
• Their framework generalizes classical coding invariants like Lee and Hamming weights, enabling precise computation of binomial moments and weight distributions.

Optimal Lee-metric anticodes are a distinguished family of submodules of finite rank in the module $R^n$ for $R=\mathbb{Z}/p^s\mathbb{Z}$, $p$ odd, $s\geq 1$, characterized by having maximal size subject to a given Lee-weight constraint. These anticodes are central objects in coding theory over rings, with their structure governed by the interplay of block-diagonal generation, support subtypes, and distributive lattice theory. Their properties underpin the development of new coding-theoretic invariants, including generalized Lee and Hamming weights, through precise combinatorial correspondences with weak compositions and Möbius inversion on poset lattices (Bariffi et al., 12 Jan 2026).

1. The Lee Metric on $\mathbb{Z}/p^s\mathbb{Z}$ and Anticode Constraints

Let $R=\mathbb{Z}/p^s\mathbb{Z}$, with $p$ an odd prime and $s\geq 1$. For $x=(x_1, \ldots, x_n)\in R^n$, the Lee weight is defined componentwise as

$$\operatorname{wt}_L(x) = \sum_{i=1}^n \min\{x_i,\,p^s-x_i\}\,.$$

The Lee distance is then $d_L(x,y) = \operatorname{wt}_L(x-y)$, generalizing the classical Lee metric to the chain ring setting.

An $R$-linear $r$-anticode $\mathcal{A}\subset R^n$ is a submodule for which the maximum Lee weight obeys $$\max\{\operatorname{wt}_L(a) : a\in \mathcal{A}\} \leq r$$. Given an $R$-submodule of rank $K$ with structure decomposition

$C\cong \bigoplus_{i=0}^{s-1} (R/p^{s-i}R)^{k_i},$

the constraint on Lee weight can be analyzed via the so-called anticode bound:

$$\max\operatorname{wt}_L(\mathcal{A}) \geq \sum_{i=0}^{s-1} k_i\,M_i,$$

where $M_i = p^i \left\lfloor p^{s-i}/2 \right\rfloor$ is the maximum Lee weight for the ideal $p^i R$ [(Bariffi et al., 12 Jan 2026), Prop. 4.13]. Equality characterizes optimal Lee-metric anticodes.

2. Explicit Characterization and Generator Structure

Optimal Lee-metric anticodes over $R^n$ are precisely the submodules whose generator matrices are block-diagonal, with blocks corresponding to the invariants of each exponent $i$:

$$A_{(k_0,\ldots,k_{s-1},n-K)} = \begin{pmatrix} I_{k_0} & 0 & \cdots & 0 & 0 \ 0 & p I_{k_1} & \cdots & 0 & 0 \ \vdots & & \ddots & \vdots & \vdots \ 0 & 0 & \cdots & p^{s-1} I_{k_{s-1}} & 0 \end{pmatrix},$$

with $K = \sum_{i=0}^{s-1} k_i$ and $n-K$ all-zero columns. Any optimal anticode is permutation equivalent to one generated by such a matrix, ensuring no off-diagonal blocks to maintain the minimal possible maximal Lee weight [(Bariffi et al., 12 Jan 2026), Thm. 2.1]. The block sizes $(k_0,\ldots,k_{s-1})$ correspond to the code's “subtype.”

3. Support Subtypes, Weak Compositions, and Family Partition

The support subtype of a code $\mathcal{A}\subset R^n$ is the tuple $a=(a_0,\ldots,a_s)$ where $a_i$ (for $i=0,\ldots,s-1$) is the number of coordinates generating $p^i R$ and $a_s$ is the number of zero coordinates. These tuples form the set $\Delta_{s+1}(n)$ of weak $(s+1)$-compositions of $n$, partitioning all optimal anticodes into finite families $\mathcal{A}_a$.

Every code in $\mathcal{A}_a$ is permutation equivalent to the canonical block-diagonal form associated with $a$. The number of permutation-equivalent copies is given by multinomial coefficients. For example, over $\mathbb{Z}/9\mathbb{Z}$ ($p=3$, $s=2$) with $n=3$ and $a=(1,1,1)$, there are six inequivalent optimal anticodes of this form [(Bariffi et al., 12 Jan 2026), Ex. 3.1].

4. Lattice Structure and Isomorphism to Weak Composition Poset

The set $\Delta_{s+1}(n)$ is partially ordered by dominance:

$$a\preceq b \iff \widehat{a}_j\leq \widehat{b}_j\;\;\forall j=0,\ldots,s,$$

where $\widehat{a}_j = a_0+\cdots+a_j$. This poset forms a finite distributive lattice. There is a bijective, order-preserving correspondence between the inclusion lattice of optimal anticodes and $(\Delta_{s+1}(n),\preceq)$ [(Bariffi et al., 12 Jan 2026), Prop. 4.22]:

$$\mathcal{A}\in\mathcal{A}_a,\, \mathcal{B}\in\mathcal{A}_b: \mathcal{A}\subset \mathcal{B} \iff a\preceq b.$$

Covering relations in the lattice correspond to minimal inclusions: $a\lessdot b$ is a cover if for some $j\in\{0,\ldots,s-1\}$, $b_j=a_j+1$, $b_{j+1}=a_{j+1}-1$, and all other components agree. Thus, the poset structure of weak compositions directly determines the hierarchy of anticode inclusions.

5. Lattice Operations and Meet/Join of Anticodes

The join and meet in the anticode lattice follow from the dominance lattice:

• For $\mathcal{A}\in\mathcal{A}_a$, $\mathcal{B}\in\mathcal{A}_b$:

$$\mathcal{A}\vee\mathcal{B} = \mathcal{A}_{a\vee b}, \quad \mathcal{A}\wedge\mathcal{B} = \mathcal{A}_{a\wedge b},$$

where coordinates of $a\vee b$ are computed by

$$c_0 = \max(\widehat{a}_0, \widehat{b}_0),\quad c_j = \max(\widehat{a}_j, \widehat{b}_j) - \widehat{c}_{j-1}\,\,\text{for }j\geq 1,$$

and similarly for meet with $\min$ [(Bariffi et al., 12 Jan 2026), Lem. 3.4]. This explicit description enables effective computation of intersections and unions within the anticode framework.

6. Coding-Theoretic Invariants: Binomial Moments, Weight Distributions, $R$-Weights

Generalizing classical weight enumerators, optimal Lee-metric anticodes parametrized by $a\in\Delta_{s+1}(n)$ enable the definition of novel coding-theoretic invariants:

• Binomial moment: For rank $K$ code $C$, define

$$B_a^{(j)}(C) = \sum_{\mathcal{A}\in\mathcal{A}_a} \#\{D\leq C\cap\mathcal{A}: \operatorname{rk}(D) = j\}.$$

• Weight distribution:

$$W_a^{(j)}(C) = \sum_{\mathcal{A}\in\mathcal{A}_a} \#\{D\leq C\cap\mathcal{A}: \operatorname{rk}(D) = j,\, D\text{ maximal in }C\cap\mathcal{A} \}.$$

Möbius inversion in $(\Delta_{s+1}(n),\preceq)$ yields explicit transforms between $B_a^{(j)}(C)$ and $W_a^{(j)}(C)$, with coefficients determined by multinomial factors and the lattice Möbius function [(Bariffi et al., 12 Jan 2026), Thm. 6.15].

For each $r=1,\ldots,K$, the $r$th $R$-weight of $C$ is given by the minimal $a$ (in a linear extension of $\preceq$) such that $\operatorname{rk}(\mathcal{A}\cap C)\geq r$ for some $\mathcal{A}\in\mathcal{A}_a$. This generalizes both the Lee and Hamming weights by varying the subclass of anticodes considered.

7. Examples and Generalizations

For $R=\mathbb{Z}/9\mathbb{Z}$, $n=3$, and $C$ generated by $G = \begin{bmatrix} 1~2~1\0~3~0\end{bmatrix}$ (rank 2), the support subtype is $(1,1,1)$. All optimal anticodes of this subtype are block-diagonal images under coordinate permutation. Computations yield $B_{(1,1,1)}^{(1)}(C)=4$ and $W_{(1,1,1)}^{(1)}(C)=4$ in accordance with Möbius inversion predictions.

Restricting to “free” anticodes recovers the classical generalized Hamming weights, thus the framework unifies and extends code invariants over finite fields and rings. This correspondence is a key application, providing tools for the analysis of code intersections and propagation, as well as insights into the combinatorial geometries underlying code families (Bariffi et al., 12 Jan 2026).

---

Summary Table: Key Properties of Optimal Lee-Metric Anticodes

Notion	Definition or Structure	Governing Set
Generator Form	Block-diagonal; diag$(I_{k_0}, pI_{k_1},…,p^{s-1}I_{k_{s-1}})$ plus $0$ columns	Subtype $(k_0,\ldots,k_{s-1})$
Support Subtype	$a=(a_0,\ldots,a_s)$ with $\sum a_i=n$	$\Delta_{s+1}(n)$
Lattice Structure	Inclusion of anticodes $\iff$ dominance in $\Delta_{s+1}(n)$	Weak composition poset
Coding Invariants	Binomial moments, weight distributions, $R$-weights	Möbius inversion

The study of optimal Lee-metric anticodes thus provides a rigorous combinatorial and algebraic foundation for generalizing distance and weight invariants in coding theory over finite chain rings, rooted in distributive lattice theory and explicit structural characterizations (Bariffi et al., 12 Jan 2026).