arXiv subjects

Daniel P. Bossaller

Publications and source records attributed to Daniel P. Bossaller.

4 recordsLinked to original sources

arXiv2025

Linear exact repair schemes for free MDS and Reed-Solomon codes over Galois rings

Codes over rings, especially over Galois rings, have been extensively studied for nearly three decades due to their similarity to linear codes over finite fields. A distributed storage system uses a linear code to encode a large file across several nodes. If one of the nodes fails, a linear exact repair scheme efficiently recovers the failed node by accessing and downloading data from the rest of the servers of the storage system. In this article, we develop a linear repair scheme for free maximum distance separable codes, which coincide with free maximum distance with respect to the rank codes over Galois rings. In particular, we give a linear repair scheme for full-length Reed-Solomon codes over a Galois ring.

cs.IT↗

arXiv2021

Ideal Extensions and Directly Infinite Algebras

Directly infinite algebras, those algebras, $E$ which have a pair of elements $x$ and $y$ where $1 = xy \neq yx$, are well known to have a sub-algebra isomorphic to $M_\infty(K)$, the set of infinite $\zplus \times \zplus$-indexed matrices which have only finitely many nonzero entries. When this sub-algebra is actually an ideal, we may analyze the algebra in terms of an extension of some algebra $A$ by $M_\infty(K)$, that is, a short exact sequence of $K$-algebras $0 \to M_\infty(K) \to E \to A \to 0$. The present article characterizes all trivial (split) extensions of $K[x,x^{-1}]$ by $M_\infty(K)$ by examining the extensions as sub-algebras of infinite matrix algebras. Furthermore, we construct an infinite family of pairwise non-isomorphic extensions $\{\mathcal T_i : i \geq 0\}$, all of which can be written as an extension $0 \to M_\infty(K) \to \mathcal T_i \to K[x,x^{-1}] \to 0$.

math.RA↗

arXiv2019

A Family of Almost Invertible Infinite Matrices

An algebraic analogue of the family of Fredholm operators is introduced for the family of row and column finite matrices, dubbed "Fredholm matrices." In addition, a measure is introduced which indicates how far a Fredholm matrix is from an invertible matrix. It is further shown that this measure respects multiplication, is invariant under perturbation by a matrix from $M_\infty(K)$, and is invariant under conjugation by an invertible row and column finite matrix.

math.RA↗

arXiv2018

On the Associativity of Infinite Matrix Multiplication

A natural definition of the product of infinite matrices mimics the usual formulation of multiplication of finite matrices with the caveat (in the absence of any sense of convergence) that the intersection of the support of each row of the first factor with the support of each column of the second factor must be finite. Multiplication is hence not completely defined, but restricted to a specific relation on infinite matrices. In order for the product of three infinite matrices $A$, $B$, and $C$ to behave in an associative manner, the middle factor, $B$, must link $A$ and $C$ in three ways: (i) $AB$ and $BC$ must both be defined; (ii) $A(BC)$ and $(AB)C$ must both be defined; and, finally, (iii) $A(BC)$ must equal $(AB)C$. In this article, these conditions are studied and are characterized in various ways by means of summability notions akin to those of formal calculus.

math.RA↗