Small-Gain Nash (SGN) Metrics

• Small-Gain Nash (SGN) is a unifying framework for block-weighted poset metrics, integrating models like Hamming, Lee, and poset metrics in finite vector spaces.
• It leverages local block transformations and poset-induced dependencies to provide precise analysis of weight distributions, isometry groups, and combinatorial structures.
• The SGN framework has practical applications in coding theory, random discrete models, and combinatorial design, offering insights into MDS codes and optimal code constructions.

Small-Gain Nash (SGN) refers to a unifying structural and geometric framework encompassing a broad set of block-weighted metric structures in finite vector spaces, with applications in coding theory, combinatorics, and random discrete models. The small-gain property characterizes metrics generated by local block transformations with poset-induced dependencies, and underlies the class of weighted poset block metrics and their generalizations. This class contains the classical Hamming, Lee, poset, pomset, block, and poset-block metrics as special cases, and supports a comprehensive analysis of weight distributions, isometry groups, code bounds, and geometric-combinatorial properties.

1. Formal Definition and Construction

Let $V = \mathbb{F}_q^{N}$ be a finite vector space decomposed into $n$ blocks, $V = \bigoplus_{i=1}^n \mathbb{F}_q^{k_i}$, where the $i$-th block has length $k_i$. Introduce a weight function $w: \mathbb{F}_q \to \mathbb{N}$, with $M_w = \max_{a \in \mathbb{F}_q} w(a)$ and $m_w = \min_{a \ne 0} w(a)$. For each block, define the induced block-weight $$w_{k_i}(x^{(i)}) = \max_{1 \leq j \leq k_i} w(x_{i,j})$$. For vectors $x \in V$, the $\pi$-support $\mathrm{supp}_\pi(x) = \{ i : x^{(i)} \neq 0 \}$ generates an order ideal $I_{P,\pi}(x)$ in a chosen poset $P = ([n], \preceq)$. The set of maximal elements $M_{P,\pi}(x)$ of $I_{P,\pi}(x)$ captures highest-level nonzero blocks.

Define the SGN (block-weighted poset) norm: $$\mathrm{wt}_{P,w, \pi}(x) = \sum_{i \in M_{P, \pi}(x)} w_{k_i}(x^{(i)}) + |I_{P, \pi}(x) \setminus M_{P, \pi}(x)| \cdot M_w$$ This yields the SGN metric: $d_{P,w, \pi}(x, y) = \mathrm{wt}_{P, w, \pi}(x-y)$ This metric class subsumes the Hamming, Lee, block, poset, poset-block, and pomset metrics for suitable choices of $P$, $\pi$, and $w$ (Ma et al., 2023, Ma et al., 2023, Shriwastva et al., 21 Jan 2025, Shriwastva et al., 2022).

2. Structural Properties and Unification

The SGN framework unifies prior metric structures via its parameterization:

• Hamming metric: $P$ antichain, $k_i=1$, $w$ is Hamming weight.
• Lee metric: $P$ antichain, $k_i=1$, $w$ is Lee weight.
• Block (error-block) metric: $P$ antichain, general $k_i$, $w$ is Hamming.
• Poset metric: $k_i=1$, general $P$, Hamming $w$.
• Weighted poset: $k_i=1$, general $P$, general $w$.
• Poset-block metric: general $P$, general $k_i$, Hamming $w$.
• Niederreiter-Rosenbloom-Tsfasman (NRT): $P$ chain, general $k_i$, Hamming $w$.
• Pomset/pomset-block: general $P$, possible Lee $w$, general $k_i$ (Ma et al., 2023, Shriwastva et al., 21 Jan 2025, Shriwastva et al., 2022).

By means of this block-weighted poset metric, geometric and combinatorial structures—balls, spheres, code parameters—can all be analyzed within a common theoretical scaffold.

3. Weight Distribution and Geometric Enumeration

The weight distribution of the SGN metric space describes the number of vectors at a given weight, generalizing classical MacWilliams-type formulas. Let $D_{i,r}$ denote the set of block-vectors in block $i$ of block-weight $r$. For each order ideal $I$ of size $i$ and $j=|\max(I)|$, compute all partitions of $r-(i-j)M_w$ among $j$ maximal blocks. The full formula for the weight distribution is: $$A_r = \sum_{i=1}^{n} \sum_{I \in \mathcal{J}_i} \sum_{j=1}^{i} \sum_{\mathbf{b} \in \mathrm{PRT}_{i - j}[r]} \left[ \sum_{\text{arrangements}} \prod_{\ell=1}^{j} |D_{i_\ell, b_\ell}| \right] q^{\sum_{s \notin I} k_s}$$ where $\mathcal{J}_i$ is the set of size-$i$ ideals and arrangements sum over all distinct orderings of the partition (Shriwastva et al., 21 Jan 2025, Shriwastva et al., 2022).

This generality enables precise counting for standard and novel metrics, including two-block, chain, and hierarchical cases. The SGN geometric structure contains all classical weight spectra as limiting or special instances.

4. Code Bounds, Perfect and MDS Codes

Within the SGN metric, code parameters obey Singleton-type and sphere-packing bounds generalizing almost all classical results. For a code $C \subseteq V$ of minimum distance $d$, the Singleton bound is: $$d_{P,w,\pi}(C) - m_w \leq \max_{J \subseteq [n], \sum_{i \in J} k_i \leq N - \log_q |C|} \sum_{i \in J} k_i \leq N - \log_q |C|$$ Codes attaining equality are maximum distance separable (MDS) in the SGN sense (Shriwastva et al., 21 Jan 2025, Ma et al., 2023, Shriwastva et al., 2022).

In the chain-poset (NRT) case with equal block sizes, MDS codes coincide exactly with the class of perfect codes: the $r$-balls around codewords partition the whole space if and only if $C$ is MDS. Moreover, there is a duality theorem: for equal-size blocks, MDS codes are in bijection (by duality) with perfect codes for the complement ideals in the dual poset (Shriwastva et al., 21 Jan 2025).

5. Isometry Groups and Symmetries

The automorphism group of the SGN metric space decomposes as a semidirect product: $$\mathrm{GL}_{w, (P, \pi)}(V) \cong \mathcal{N} \rtimes \mathrm{Aut}(P, \pi)$$ where $\mathcal{N}$ consists of upper-triangular block matrices respecting the poset structure, and $\mathrm{Aut}(P, \pi)$ permutes blocks among automorphisms of the poset with fixed block sizes. For classical metrics:

• Hamming: the isometry group is $(\mathbb{F}_q^*)^n \rtimes S_n$.
• Lee: $\{\pm 1\}^n \rtimes S_n$.
• General SGN: block-upper-triangular invertibles with block-permuting automorphisms (Ma et al., 2023).

This structure ensures that SGN metric balls and spheres are fully symmetric under these isometry actions, and that the associated combinatorial geometry is invariant under their action.

6. Geometric and Combinatorial Insights

The geometry of the SGN (block-weighted poset) space is governed by the interplay of block structure, poset dependencies, and symbol-weight functions. Balls of fixed radius correspond to unions of affine block subspaces, viewed as "plateaus" determined by the collection of order ideals, with maximal blocks contributing exact local weights and non-maximal blocks incurring full cost $M_w$. The combinatorial structure is thus an intricate tiling by unions of axis-parallel "faces" of varying heights, interpolating between Hamming cubes, Lee octahedra, and NRT block simplices.

This geometric insight underlies packing and covering arguments, code construction, and direct computation of packing/covering radii. Slices of the SGN geometry, parameterized by $w$ and $P$, yield all previously known "small-gain" or block-weighted metric models as described in (Ma et al., 2023, Shriwastva et al., 2022, Ma et al., 2023).

7. Applications and Further Directions

SGN metrics are foundational in the study of error-correcting codes, random discrete geometric models, and combinatorial design. The universality of the construction supports new code families, explicit formulae for weight distributions, and duality theory for MDS codes in highly structured spaces (Shriwastva et al., 21 Jan 2025). SGN geometry has also been leveraged in phase transition analysis of block-weighted random maps, where the shifts in geometric regime correspond to varying block-weight parameters—a connection illuminating the breadth of the small-gain framework (Fleurat et al., 2023).

Further research develops geometric invariants, enumerative theory, optimal code constructions in new SGN instances, as well as probabilistic and combinatorial applications where small-gain decompositions yield tractable analytic results.

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Key references:

• "Codes with Weighted Poset Block Metrics" (Ma et al., 2023)
• "Groups of linear isometries on weighted poset block spaces" (Ma et al., 2023)
• "Weight Distribution of the Weighted Coordinates Poset Block Space and Singleton Bound" (Shriwastva et al., 21 Jan 2025)
• "Weighted Coordinates Poset Block Codes" (Shriwastva et al., 2022)
• "A phase transition in block-weighted random maps" (Fleurat et al., 2023)