Cauchy MDS Matrices over Galois Rings

• Cauchy MDS matrices over Galois rings are defined using Teichmüller sets and nilpotent elements to guarantee that every square submatrix is invertible.
• They are constructed in distinct families (Type-I and Type-II) by selecting disjoint tuples or employing nilpotent shifts to optimize computational efficiency.
• Frobenius automorphisms and unit multiplications enable the generation of numerous equivalent matrices while preserving the MDS property across different ring isomorphisms.

A Cauchy MDS matrix over a Galois ring is an analogue of the classical Cauchy matrix defined over a finite field, possessing the maximum distance separable (MDS) property: every square submatrix is invertible. The construction of such matrices over Galois rings expands the algebraic infrastructure beyond fields, introducing additional parameters and leveraging the structure of nilpotent elements and Frobenius automorphisms to produce diverse explicit families of MDS matrices (Ali et al., 22 Dec 2025).

1. Galois Rings: Structure and Key Components

Let $p$ be any prime, and $s, m$ be positive integers. A Galois ring $GR(p^s, p^{sm})$ is defined as the quotient $\mathbb{Z}_{p^s}[x]/(f(x))$, where $f(x)$ is a monic basic irreducible polynomial of degree $m$ (i.e., its reduction mod $p$ is irreducible in $\mathbb{F}_p[x]$). This ring has characteristic $p^s$ and cardinality $p^{sm}$. The canonical projection $\mathbb{Z}_{p^s}\to\mathbb{Z}_p$ extends to $GR(p^s,p^{sm})$, inducing a reduction map to the finite field $F_{p^m}$.

A fundamental object in the construction is the Teichmüller set $\tau = \{0,1,\xi,\xi^2, \dots, \xi^{p^m-2}\}$, where $\xi$ is a Teichmüller generator of order $p^m - 1$, constructed from a basic primitive polynomial $h(x)$ dividing $x^{p^m-1}-1$. Every $t \in GR(p^s, p^{sm})$ possesses a unique $p$-adic expansion:

$$t = t_0 + p t_1 + \cdots + p^{s-1} t_{s-1}, \quad t_i \in \tau.$$

Nilpotent elements in this context are those in the maximal ideal $(p)$—that is, all multiples of $p$—and form the set $\mathcal{N}$. The units $U$ of the ring are those elements whose reduction modulo $p$ is nonzero.

2. Construction of Cauchy MDS Matrices

Type-I (First Kind)

Choose two disjoint $k$-tuples of nonzero, distinct Teichmüller elements $x_1, \dots, x_k$ and $y_1, \dots, y_k$ from $\tau \setminus \{0\}$. The Type-I Cauchy matrix $A$ is given by

$A_{ij} = \frac{1}{x_i - y_j}.$

The determinantal formula,

$$\det A = \frac{ \prod_{1 \leq j < i \leq k} (x_i-x_j)(y_j-y_i) }{ \prod_{1 \leq i,j \leq k} (x_i - y_j) } \in U,$$

follows from the fact that differences of Teichmüller elements are always units. Every square submatrix of $A$ is again Cauchy; hence, $A$ is MDS.

Type-II (Second Kind) and Nilpotent Reduction

For $p\ne 2$, consider the subset $$\tau' = \{0,1,\xi,\dots, \xi^{\lceil (p^m-2)/2 \rceil} \}$$. If $x_i, y_j \in \tau' \setminus \{0\}$ are distinct, the matrix

$A_{ij} = \frac{1}{x_i + y_j}$

is MDS, again by invertibility via units.

A critical reduction arises from nilpotent elements. Fix any nilpotent $\ell \in \mathcal{N}$ and set $y_j = x_j + \ell$ with $x_1,\dots,x_k \in \tau' \setminus \{0\}$, constructing the symmetric matrix

$A_{ij} = \frac{1}{x_i + x_j + \ell}.$

For $i\ne j$, $x_i + x_j$ is a unit. For $i=j$, $2x_i + \ell$ remains a unit since $2\in U$ for $p\ne 2$. The number of distinct denominators reduces from $k^2$ to $\frac{k(k+1)}{2}$, an improvement for memory or implementation efficiency.

Table: Types of Cauchy MDS Matrices over $GR(p^s, p^{sm})$

Matrix Type	Formulation	Prerequisites
Type-I	$1/(x_i - y_j)$	Disjoint $x_i, y_j$ in $\tau\setminus\{0\}$
Type-II	$1/(x_i + y_j)$ or $1/(x_i + x_j + \ell)$	$p\ne 2$, $x_i, y_j$ in $\tau'$, $\ell\in\mathcal{N}$

3. Necessary Conditions for the MDS Property

The guaranteed invertibility of all minors (the MDS property) in these constructions depends fundamentally on the properties of the Teichmüller parameters:

• Differences $x_i-x_j$, sums $x_i+x_j$ (Type-II), and shifts by nilpotents must remain units in $GR(p^s, p^{sm})$.
• For Type-II sums, $p\ne 2$ is required, except in the characteristic-$2$ case where the sum of distinct Teichmüller units remains a unit.
• In odd characteristic, restrictions like $\xi^{\sigma_i-\eta_j} \ne -1$ prevent denominators from becoming zero.

4. Frobenius Automorphisms and Distinct MDS-Preserving Maps

Elements $t \in GR(p^s, p^{sm})$ admit a Teichmüller $p$-adic decomposition. The Frobenius automorphism $\sigma$ acts as

$\sigma(t) = \sum_{\ell=0}^{s-1} p^\ell t_\ell^p.$

This automorphism is of order $m$ and fixes $\mathbb{Z}_{p^s}$.

Generalizing, consider the extension ring $GR(p^s, p^{sm\ell})$. Define $\varphi^r$ by

$$\varphi^r\left(\sum_{i=0}^{\ell-1} a_i \xi^i\right) = \sum_{i=0}^{\ell-1} a_i \xi^{i p^{m r}}, \quad 0 \leq r < \ell.$$

Given a Cauchy-MDS matrix $A$ over $GR(p^s,p^{sm\ell})$, define new matrices by applying the automorphism to each entry denominator or to the entries directly: $A'_{ij} = \varphi^r(a_{ij})$.

By closure under automorphisms, $\det A'$ is a unit if and only if $\det A$ is a unit; thus $\varphi^r$ preserves the MDS property. Compounding Frobenius automorphisms with multiplication by units $c \in U$ yields $p^{(s-1)m}(p^m-1)$ distinct bijections $f_{r,c}: t \mapsto \varphi^r(t) \cdot c$, generating the corresponding number of distinct MDS matrices.

5. Automorphism and Isomorphism-Based Families

Two key methods generalize the construction of new Cauchy MDS matrices:

A. Frobenius Automorphisms within a Single Galois Ring

The maps $f_{r,c}: t \mapsto \sigma^r(t) \cdot c$ (for $0 \leq r < m$, $c \in U$) can be applied entrywise to a Cauchy-MDS matrix $A$ to produce $A'$ with $\det A' = c^k \sigma^r(\det A)$, which remains a unit.

B. Isomorphisms Between Galois Rings Defined by Different Primitive Polynomials

Let $h_1(x), h_2(x)$ be basic irreducible polynomials of degree $m$ with roots $\eta_1, \eta_2$. For each $s_u = e p^i$ with $1 \leq e \leq p^m-2$, $\gcd(e, p^m-1)=1$, $0\leq i<m$, $\eta_2^{s_u}$ is also a root of $h_2$. Define

$$f_{s_u,c}\left( \sum_{j=0}^{m-1} a_j \eta_1^j \right) = c \sum_{j=0}^{m-1} a_j (\eta_2)^{j s_u}.$$

Each $f_{s_u,c}$ is a bijection that preserves matrix operations up to multiplication by a unit, thus mapping Cauchy-MDS matrices between the corresponding Galois rings.

6. Implementation and Computational Remarks

Utilizing $GR(p^s,p^{sm})$ rather than a finite field $F_{p^m}$ introduces an additional "p-adic" parameter $s$, facilitating flexibility for hardware or software that supports arithmetic in $\mathbb{Z}_{p^s}$. The comprehensive set of Teichmüller representatives allows systematic selection of units, simplifying the generalization of classical Cauchy-MDS matrices.

Reduction via nilpotent shifts in Type-II constructions leads to storage and computational efficiencies in implementations due to the reduced number of distinct matrix entries. The application of Frobenius automorphisms $\sigma^r$ and unit multiplications $c$ allows generation of $p^{(s-1)m}(p^m-1)$ distinct, equivalent-diffusing MDS matrices from any given Cauchy-MDS seed.

Isomorphisms between Galois ring realizations with different primitive polynomials extend these families further, enabling large-scale generation of Cauchy-MDS matrices across various Galois ring incarnations.

In practice, the explicit construction procedure involves:

1. Selecting $p,s,m$ and constructing $\xi$ via a suitable primitive polynomial.
2. Choosing $x_i, y_j$ from the Teichmüller set (with or without nilpotent shift $\ell$).
3. Composing $A_{ij} = 1/(x_i - y_j)$ or $A_{ij} = 1/(x_i + x_j + \ell)$ as appropriate.
4. Verifying $p\ne 2$ constraint or applying the characteristic-$2$ case for Type-II.
5. Optionally applying $f_{r,c}$ (Frobenius automorphism and multiplication) for further matrix generation.

These procedures are explicit and implementable in any computer algebra system supporting arithmetic in $\mathbb{Z}_{p^s}[x]/(f(x))$ (Ali et al., 22 Dec 2025).