--- title: Cauchy MDS Matrices over Galois Rings url: https://www.emergentmind.com/topics/cauchy-mds-matrices-over-galois-rings type: topic --- # Cauchy MDS Matrices over Galois Rings 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 [2512.19306]. ## 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