arXiv subjects

Henk D. L. Hollmann

Publications and source records attributed to Henk D. L. Hollmann.

At least 19 recordsLinked to original sources

arXiv2026

Recovery Models for Linear Batch Codes

Various types of recovery algorithms for batch codes have been considered previously, including asynchronous recovery and recovery algorithms arising from Almost Affinely Disjoint (AAD) families. We initiate a systematic investigation of linear batch codes equipped with recovery algorithms. We introduce online and strongly online batch codes, investigate their relations with asynchronous batch codes, and introduce (m,L)-strongly online batch codes, which provide particularly simple recovery algorithms. We then study simple batch codes associated with arbitrary bipartite graphs, obtaining graph-theoretic conditions for such codes and generalizing several known results.

cs.IT↗

arXiv2026

Non-binary LDPC codes for Data Storage

In modern data storage systems, non-binary LDPC codes for recovering from disk failures are increasingly considered strong competitors to MDS codes such as Reed-Solomon codes. Since disk failures can be modeled as erasures, we analyze non-binary LDPC codes over a $q$-ary field in the $q$-ary erasure channel, relative to MDS codes. Our focus is on non-binary LDPC codes whose parity-check matrix is obtained by replacing the non-zero entries of a binary base matrix by elements of a $q$-ary finite field. For such LDPC codes, we introduce the notion of ultimate distance, which upper-bounds their minimum distance. We derive a random-coding bound on the number of non-correctable erasure patterns for the Gallager ensemble of regular non-binary LDPC codes under maximum-likelihood decoding. An algorithm for finding the ultimate distance is presented. A low-complexity algorithm for searching for the minimum distance of the non-binary LDPC code is proposed. Finally, we construct examples of non-binary LDPC codes achieving the ultimate distance.

cs.IT↗

arXiv2026

On the Minimum Length of Functional Batch Codes with Small Recovery Sets

Batch codes are of potential use for load balancing and private information retrieval in distributed data storage systems. Recently, a special case of batch codes, termed functional batch codes, was proposed in the literature. In functional batch codes, users can query linear combinations of the information symbols, and not only the information symbols themselves, as is the case for standard batch codes. In this work, we consider linear functional batch codes with the additional property that every query is answered by using only a small number of coded symbols. We derive bounds on the minimum length of such codes, and evaluate the results by numerical computations.

cs.IT↗

arXiv2026

The permutation automorphism groups of irreducible cyclic codes

The study of permutation automorphism groups of cyclic codes is a central topic in algebraic coding theory. A cyclic code over $\mathbb{F}_q$ is called irreducible if its check polynomial is irreducible over $\mathbb{F}_q$. Such a code is standard if its permutation automorphism group is equal to the group generated by the cyclic shift and the Frobenius automorphism, and non-standard otherwise. In this paper, we give a complete classification of all non-standard non-degenerate irreducible cyclic codes, using the classification of finite simple groups. Our result shows that, apart from a small number of explicit exceptional families and their descendants under certain secondary constructions, every non-degenerate irreducible cyclic code is standard, and up to four explicit exceptions, every degenerate cyclic code is non-standard. This classification has several consequences. First, it yields a general description of non-standard linear recurring sequence subgroups, extending the earlier work of Brison and Nogueira; secondly it establishes the Schmidt-White conjecture for all non-standard irreducible cyclic codes. Moreover, our results provide strong evidence in support of the conjecture of Berger and Charpin that almost all cyclic codes are standard.

math.CO↗

arXiv2024

PIR Codes, Unequal-Data-Demand Codes, and the Griesmer Bound

Unequal Error-Protecting (UEP) codes are error-correcting (EC) codes designed to protect some parts of the encoded data better than other parts. Here, we introduce a similar generalization of PIR codes that we call Unequal-Data-Demand (UDD) PIR codes. These codes are PIR-type codes designed for the scenario where some parts of the encoded data are in higher demand than other parts. We generalize various results for PIR codes to UDD codes. Our main contribution is a new approach to the Griesmer bound for linear EC codes involving an Integer Linear Programming (ILP) problem that generalizes to linear UEP codes and linear UDD PIR codes.

cs.IT↗

arXiv2023

On some batch code properties of the simplex code

The binary $k$-dimensional simplex code is known to be a $2^{k-1}$-batch code and is conjectured to be a $2^{k-1}$-functional batch code. Here, we offer a simple, constructive proof of a result that is "in between" these two properties. Our approach is to relate these properties to certain (old and new) additive problems in finite abelian groups. We also formulate a conjecture for finite abelian groups that generalizes the above-mentioned conjecture.

cs.IT↗

arXiv2022

Optimal possibly nonlinear 3-PIR codes of small size

First, we state a generalization of the minimum-distance bound for PIR codes. Then we describe a construction for linear PIR codes using packing designs and use it to construct some new 5-PIR codes. Finally, we show that no encoder (linear or nonlinear) for the binary $r$-th order Hamming code produces a 3-PIR code except when $r=2$. We use these results to determine the smallest length of a binary (possibly nonlinear) 3-PIR code of combinatorial dimension up to~6. A binary 3-PIR code of length 11 and size $2^7$ is necessarily nonlinear, and we pose the existence of such a code as an open problem.

cs.IT↗

arXiv2022

Non-standard linear recurring sequence subgroups and automorphisms of irreducible cyclic codes

Let $\cU$ be the multiplicative group of order~$n$ in the splitting field $\bbF_{q^m}$ of $x^n-1$ over the finite field $\bbF_q$. Any map of the form $x\rightarrow cx^t$ with $c\in \cU$ and $t=q^i$, $0\leq i<m$, is $\bbF_q$-linear on~$\bbF_{q^m}$ and fixes $\cU$ set-wise; maps of this type will be called {\em standard\/}. Occasionally there are other, {\em non-standard\/} $\bbF_q$-linear maps on~$\bbF_{q^m}$ fixing $\cU$ set-wise, and in that case we say that the pair $(n, q)$ is {\em non-standard\/}. We show that an irreducible cyclic code of length~$n$ over $\bbF_q$ has ``extra'' permutation automorphisms (others than the {\em standard\/} permutations generated by the cyclic shift and the Frobenius mapping that every such code has) precisely when the pair $(n, q)$ is non-standard; we refer to such irreducible cyclic codes as {\em non-standard\/} or {\em NSIC-codes\/}. In addition, we relate these concepts to that of a non-standard linear recurring sequence subgroup as investigated in a sequence of papers by Brison and Nogueira. We present several families of NSIC-codes, and two constructions called ``lifting'' and ``extension'' to create new NSIC-codes from existing ones. We show that all NSIC-codes of dimension two can be obtained in this way, thus completing the classification for this case started by Brison and Nogueira.

cs.IT↗

arXiv2021

Some basic results on finite linear recurring sequence subgroups

An $f$-subgroup is a linear recurring sequence subgroup, a multiplicative subgroup of a field whose elements can be generated (without repetition) by a linear recurrence relation, with characteristic polynomial $f$. It is called non-standard if it can be generated in a non-cyclic way (that is, not in the order $α^i, α^{i+1}, α^{i+2} \ldots$ for a zero $α$ of $f$), and standard otherwise. We will show that a finite $f$-subgroup is necessarily generated by a subset of the zeros of $f$. We use this result to improve on a recent theorem of Brison and Nogueira. A old question by Brison and Nogueira asks if there exist automatically non-standard $f$-subgroups, $f$-subgroups that cannot be generated by a zero of $f$. We answer that question affirmatively by constructing infinitely many examples.

math.CO↗

arXiv2018

The shift bound for abelian codes and generalizations of the Donoho-Stark uncertainty principle

Let $G$ be a finite abelian group. If $f: G\rightarrow \bC$ is a nonzero function with Fourier transform $\hf$, the Donoho-Stark uncertainty principle states that $|\supp(f)||\supp(\hf)|\geq |G|$. The purpose of this paper is twofold. First, we present the shift bound for abelian codes with a streamlined proof. Second, we use the shifting technique to prove a generalization and a sharpening of the Donoho-Stark uncertainty principle. In particular, the sharpened uncertainty principle states, with notation above, that $|\supp(f)||\supp(\hf)|\geq |G|+|\supp(f)|-|H(\supp(f))|$, where $H(\supp(f))$ is the stabilizer of $\supp(f)$ in $G$.

math.CO↗

arXiv2018

A Multi-layer Recursive Residue Number System

We present a method to increase the dynamical range of a Residue Number System (RNS) by adding virtual RNS layers on top of the original RNS, where the required modular arithmetic for a modulus on any non-bottom layer is implemented by means of an RNS Montgomery multiplication algorithm that uses the RNS on the layer below. As a result, the actual arithmetic is deferred to the bottom layer. The multiplication algorithm that we use is based on an algorithm by Bajard and Imbert, extended to work with pseudo-residues (remainders with a larger range than the modulus). The resulting Recursive Residue Number System (RRNS) can be used to implement modular addition, multiplication, and multiply-and-accumulate for very large (2000+ bits) moduli, using only modular operations for small (for example 8-bits) moduli. A hardware implementation of this method allows for massive parallelization. Our method can be applied in cryptographic algorithms such as RSA to realize modular exponentiation with a large (2048-bit, or even 4096-bit) modulus. Due to the use of full RNS Montgomery algorithms, the system does not involve any carries, therefore cryptographic attacks that exploit carries cannot be applied.

cs.CR↗

arXiv2014

Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields

A maximal minor $M$ of the Laplacian of an $n$-vertex Eulerian digraph $Γ$ gives rise to a finite group $\mathbb{Z}^{n-1}/\mathbb{Z}^{n-1}M$ known as the sandpile (or critical) group $S(Γ)$ of $Γ$. We determine $S(Γ)$ of the generalized de Bruijn graphs $Γ=\mathrm{DB}(n,d)$ with vertices $0,\dots,n-1$ and arcs $(i,di+k)$ for $0\leq i\leq n-1$ and $0\leq k\leq d-1$, and closely related generalized Kautz graphs, extending and completing earlier results for the classical de Bruijn and Kautz graphs. Moreover, for a prime $p$ and an $n$-cycle permutation matrix $X\in\mathrm{GL}_n(p)$ we show that $S(\mathrm{DB}(n,p))$ is isomorphic to the quotient by $\langle X\rangle$ of the centralizer of $X$ in $\mathrm{PGL}_n(p)$. This offers an explanation for the coincidence of numerical data in sequences A027362 and A003473 of the OEIS, and allows one to speculate upon a possibility to construct normal bases in the finite field $\mathbb{F}_{p^n}$ from spanning trees in $\mathrm{DB}(n,p)$.

math.CO↗

arXiv2013

Critical groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields: an extended abstract

We determine the critical groups of the generalized de Bruijn graphs DB$(n,d)$ and generalized Kautz graphs Kautz$(n,d)$, thus extending and completing earlier results for the classical de Bruijn and Kautz graphs. Moreover, for a prime $p$ the critical groups of DB$(n,p)$ are shown to be in close correspondence with groups of $n\times n$ circulant matrices over $\mathbb{F}_p$, which explains numerical data in [OEIS:A027362], and suggests the possibility to construct normal bases in $\mathbb{F}_{p^n}$ from spanning trees in DB$(n,p)$.

math.CO↗

arXiv2013

Characterizations and construction methods for linear functional-repair storage codes

We present a precise characterization of linear functional-repair storage codes in terms of {\em admissible states/}, with each state made up from a collection of vector spaces over some fixed finite field. To illustrate the usefulness of our characterization, we provide several applications. We first describe a simple construction of functional-repair storage codes for a family of code parameters meeting the cutset bound outside the MBR and MSR points; these codes are conjectured to have optimal rate with respect to their repair locality. Then, we employ our characterization to develop a construction method to obtain functional repair codes for given parameters using symmetry groups, which can be used both to find new codes and to improve known ones. As an example of the latter use, we describe a beautiful functional-repair storage code that was found by this method, with parameters belonging to the family investigated earlier, which can be specified in terms of only eight different vector spaces.

cs.IT↗

arXiv2013

Locally Repairable Codes with Multiple Repair Alternatives

Distributed storage systems need to store data redundantly in order to provide some fault-tolerance and guarantee system reliability. Different coding techniques have been proposed to provide the required redundancy more efficiently than traditional replication schemes. However, compared to replication, coding techniques are less efficient for repairing lost redundancy, as they require retrieval of larger amounts of data from larger subsets of storage nodes. To mitigate these problems, several recent works have presented locally repairable codes designed to minimize the repair traffic and the number of nodes involved per repair. Unfortunately, existing methods often lead to codes where there is only one subset of nodes able to repair a piece of lost data, limiting the local repairability to the availability of the nodes in this subset. In this paper, we present a new family of locally repairable codes that allows different trade-offs between the number of contacted nodes per repair, and the number of different subsets of nodes that enable this repair. We show that slightly increasing the number of contacted nodes per repair allows to have repair alternatives, which in turn increases the probability of being able to perform efficient repairs. Finally, we present pg-BLRC, an explicit construction of locally repairable codes with multiple repair alternatives, constructed from partial geometries, in particular from Generalized Quadrangles. We show how these codes can achieve practical lengths and high rates, while requiring a small number of nodes per repair, and providing multiple repair alternatives.

cs.IT↗

arXiv2013

Storage codes -- coding rate and repair locality

The {\em repair locality} of a distributed storage code is the maximum number of nodes that ever needs to be contacted during the repair of a failed node. Having small repair locality is desirable, since it is proportional to the number of disk accesses during repair. However, recent publications show that small repair locality comes with a penalty in terms of code distance or storage overhead if exact repair is required. Here, we first review some of the main results on storage codes under various repair regimes and discuss the recent work on possible (information-theoretical) trade-offs between repair locality and other code parameters like storage overhead and code distance, under the exact repair regime. Then we present some new information theoretical lower bounds on the storage overhead as a function of the repair locality, valid for all common coding and repair models. In particular, we show that if each of the $n$ nodes in a distributed storage system has storage capacity $\ga$ and if, at any time, a failed node can be {\em functionally} repaired by contacting {\em some} set of $r$ nodes (which may depend on the actual state of the system) and downloading an amount $\gb$ of data from each, then in the extreme cases where $\ga=\gb$ or $\ga = r\gb$, the maximal coding rate is at most $r/(r+1)$ or 1/2, respectively (that is, the excess storage overhead is at least $1/r$ or 1, respectively).

cs.IT↗

arXiv2008

Nonstandard linear recurring sequence subgroups in finite fields and automorphisms of cyclic codes

Let $q=p^r$ be a prime power, and let $f(x)=x^m-\gs_{m-1}x^{m-1}- >...-\gs_1x-\gs_0$ be an irreducible polynomial over the finite field $\GF(q)$ of size $q$. A zero $ξ$ of $f$ is called {\em nonstandard (of degree $m$) over $\GF(q)$} if the recurrence relation $u_m=\gs_{m-1}u_{m-1} + ... + \gs_1u_1+\gs_0u_0$ with characteristic polynomial $f$ can generate the powers of $ξ$ in a nontrivial way, that is, with $u_0=1$ and $f(u_1)\neq 0$. In 2003, Brison and Nogueira asked for a characterisation of all nonstandard cases in the case $m=2$, and solved this problem for $q$ a prime, and later for $q=p^r$ with $r\leq4$. In this paper, we first show that classifying nonstandard finite field elements is equivalent to classifying those cyclic codes over $\GF(q)$ generated by a single zero that posses extra permutation automorphisms. Apart from two sporadic examples of degree 11 over $\GF(2)$ and of degree 5 over $\GF(3)$, related to the Golay codes, there exist two classes of examples of nonstandard finite field elements. One of these classes (type I) involves irreducible polynomials $f$ of the form $f(x)=x^m-f_0$, and is well-understood. The other class (type II) can be obtained from a primitive element in some subfield by a process that we call extension and lifting. We will use the known classification of the subgroups of $\PGL(2,q)$ in combination with a recent result by Brison and Nogueira to show that a nonstandard element of degree two over $\GF(q)$ necessarily is of type I or type II, thus solving completely the classification problem for the case $m=2$.

cs.IT↗

arXiv2008

Proofs of two conjectures on ternary weakly regular bent functions

We study ternary monomial functions of the form $f(x)=\Tr_n(ax^d)$, where $x\in \Ff_{3^n}$ and $\Tr_n: \Ff_{3^n}\to \Ff_3$ is the absolute trace function. Using a lemma of Hou \cite{hou}, Stickelberger's theorem on Gauss sums, and certain ternary weight inequalities, we show that certain ternary monomial functions arising from \cite{hk1} are weakly regular bent, settling a conjecture of Helleseth and Kholosha \cite{hk1}. We also prove that the Coulter-Matthews bent functions are weakly regular.

math.CO↗