# Henk D.L. HollmannUniversity of Tartu · Institute of Computer Science

Henk D.L. Hollmann

PhD

## About

98

Publications

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Introduction

My current research interests are algebraic methods to investigate various batch-type and other codes. I am also interested in automorphisms of cyclic codes and linear recurring sequence subgroups.

Additional affiliations

January 2017 - August 2017

January 2015 - present

December 2011 - December 2014

## Publications

Publications (98)

We study quadratic residue difference sets, GMW difference sets, and difference sets arising from monomial hyperovals, all of which are (2d−1, 2d−1−1, 2d−2−1) cyclic difference sets in the multiplicative group of the finite field 2d of 2d elements, with d⩾2. We show that, except for a few cases with small d, these difference sets are all pairwise i...

Binary m-sequences are widely applied in navigation, radar, and communication systems because of their nice autocorrelation and cross-correlation properties. In this paper, we consider the cross-correlation between a binary m-sequence of length 2m−1 and a decimation of that sequence by an integer t. We will be interested in the number of values att...

If C is a q-ary code of length n and a and b are two codewords, then c is called a descendant of a and b if c(i) is an element of {a(i), b(i)} for i = 1, ..., n. We are interested in codes C with the property that, given any descendant c, one can always identify at least one of the "parent" codewords in C. We study bounds on F(n, q), the maximal ca...

A new N = 2n fast Fourier transform algorithm is presented, which has fewer multiplications and additions than radix 2n, n = 1, 2, 3 algorithms, has the same number of multiplications as the Raderi-Brenner algorithm, but much fewer additions, and is numerically better conditioned, and is performed `in place' by a repetitive use of a `butterfly'-typ...

We present a new technique to construct sliding-block modulation
codes with a small decoding window. Our method, which involves both
state splitting and look-ahead encoding, crucially depends on a new
“local” construction method for bounded-delay codes. We
apply our method to construct several new codes, all with a smaller
decoding window than prev...

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\/}. Oc...

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. W...

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, where the relation has characteristic polynomial f. It is called non-standard if it can be generated in a non-cyclic way (that is, not in the order αi,αi+1,αi+2… for a zero...

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 $\alpha^i, \alpha^{i+1}, \alpha^{i+2}...

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...

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...

The repair locality of a storage code is the maximum number of nodes that may 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 required during a node repair, which for certain applications seems to be the main bottleneck. However, recent publications s...

Dans cet article, la borne de- Rao-Wilson [1], ainsi que le dual du théorème de Lloyd, sont généralisés aux t-design s à points répétés dans les schémas d'association Q-polynomiaux . La démonstration utilise une généralisation d 'un résultat de Connor [5] pour les 2-designs classiques. De plus, on donne une nouvelle démonstration de l'inégalité de...

A maximal minor $M$ of the Laplacian of an $n$-vertex Eulerian digraph
$\Gamma$ gives rise to a finite group $\mathbb{Z}^{n-1}/\mathbb{Z}^{n-1}M$
known as the sandpile (or critical) group $S(\Gamma)$ of $\Gamma$. We determine
$S(\Gamma)$ of the generalized de Bruijn graphs $\Gamma=\mathrm{DB}(n,d)$ with
vertices $0,\dots,n-1$ and arcs $(i,di+k)$ fo...

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...

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-re...

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 repa...

The 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 te...

In this paper, we study ternary monomial functions of the form f ( x ) = Tr n ( axd ), where x isin BBF <sub>3</sub> n and Trn : BBF <sub>3</sub> n rarr BBF <sub>3</sub> is the absolute trace function. Using a lemma of Hou, Stickelberger's theorem on Gauss sums, and certain ternary weight inequalities, we show that certain ternary monomial function...

In 2007, Martinian and Trott presented codes for correcting a burst of erasures with a minimum decoding delay. Their construction employs $[n,k]$ codes that can correct any burst of erasures (including wrap-around bursts) of length $n-k$ . They raised the question if such $[n,k]$ codes exist for all integers $k$ and $n$ with $1leq kleq n$ and all f...

A white-box implementation of a block cipher is a software implementation from which it is difficult for an attacker to extract
the cryptographic key. Chow et al. published white-box implementations for AES and DES. These implementations are based on
ideas that can be used to derive white-box implementations for other block ciphers as well. In part...

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 $\xi$ 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...

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal,...

We give recursive constructions, valid for any field, of [n,k] codes capable of correcting a (wrap-around) burst of n - k erasures.

Recently there has been interest in the construction of small parity-check sets for iterative decoding of the Hamming code with the property that each uncorrectable (or stopping) set of size three is the support of a codeword and hence uncorrectable anyway. Here we reformulate and generalize the problem and improve on this construction. We show tha...

A generic (r, m)-erasure correcting set generates for each binary linear code of codimension r a collection of parity check equations that enables iterative decoding of all potentially correctable erasure patterns of size at most m. As we have shown earlier, such a set essentially is just a parity check collection with this property for the Hamming...

When Jack van Lint was appointed as full professor at the Eindhoven University of Technology at the age of 26 he combined a PhD in number theory with a very open scientific mind. It took a sabbatical visit to Bell Laboratories in 1966 to make him understand that a new and fascinating field of applied mathematics was emerging: discrete mathematics....

We consider a simple transformation (coding) of an iid source called a bit-shift channel. This simple transformation occurs naturally in magnetic or optical data storage. The resulting process is not Markov of any order. We discuss methods of computing the entropy of the transformed process, and study some of its properties.

A generic (r,m)-erasure correcting set is a collection of vectors in F<sub>2</sub> <sup>r</sup> which can be used to generate, for each binary linear code of codimension r, a collection of parity check equations that enables iterative decoding of all correctable erasure patterns of size at most m. That is to say, the only stopping sets of size at m...

The action of PGL(2,2 m ) on the set of exterior lines to a nonsingular conic in PG(2,2 m ) affords an association scheme, which was shown to be pseudocyclic in [(*) H. D. L. Hollmann, Association schemes, Master thesis, Eindhoven University of Technology, 1982]. It was further conjectured in [(*)] that the orbital scheme of PΓL(2,2 m ) on the set...

A generic $(r,m)$-erasure correcting set is a collection of vectors in $\bF_2^r$ which can be used to generate, for each binary linear code of codimension $r$, a collection of parity check equations that enables iterative decoding of all correctable erasure patterns of size at most $m$. That is to say, the only stopping sets of size at most $m$ for...

Abstract The action of PGL(2, 2^m) is also pseudocyclic if m is an odd prime. We confirm this conjecture in this paper. As a by-product, we obtain a class of Latin square type strongly regular graphs on nonprime-power,number,of points. © 2005 Elsevier Inc. All rights reserved. Keywords: Association scheme; Conic; Dickcon polynomial; Fusion scheme;...

The action of $PGL(2,2^m)$ on the set of exterior lines to a nonsingular conic in $PG(2,2^m)$ affords an association scheme, which was shown to be pseudocyclic in Hollmann's thesis in 1982. It was further conjectured in Hollmann's thesis that the orbital scheme of $P\Gamma L(2,2^m)$ on the set of exterior lines to a nonsingular conic in $PG(2,2^m)$...

The group $PGL(2,q)$ has an embedding into $PGL(3,q)$ such that it acts as the group fixing a nonsingular conic in $PG(2,q)$. This action affords a coherent configuration $R(q)$ on the set $L(q)$ of non-tangent lines of the conic. We show that the relations can be described by using the cross-ratio. Our results imply that the restrictions $R_{+}(q)...

In order to prevent ground bounce, Automatic Test Pattern Generation (ATPG) algorithms for wire interconnects have recently been extended with the capability to restrict the maximal Hamming distance between any two consecutive test patterns to a user-defined integer, referred to as the Simultaneously-Switching Outputs Limit (SSOL). The conventional...

We construct a class of permutation polynomials of $\bF_{2^m}$ that are
closely related to Dickson polynomials.

A recent publication introduced a Visual Crypto (VC) system, based on the polarisation of light. This VC system has goodresolution, contrast and colour properties.Mathematically, the VC system is described by the XOR operation (modulo two addition). In this paper we investigate Threshold Visual Secret Sharing schemes associated to XOR-based VC syst...

We construct a class of permutation polynomials of $\bF_{2^m}$ that are closely related to Dickson polynomials.

We introduce Kloosterman polynomials over , and use these polynomials to prove three identities involving Kloosterman sums over .

We consider lists of distinct q-ary addresses of length n. We wish that any b consecutive addresses in such a list agree in many positions. We give upper bounds on what can be achieved. Moreover, for each q and n, we give explicit constructions of address lists, among which is the conventional q-ary reflected Gray code, that attain these bounds for...

In this article, the computation of the entropy rate H(y) of a binary-valued stochastic process (Y<sub>1</sub>, Y<sub>2</sub>,...) which is a function of a stationary, time-invariant and irreducible Markov chain (X<sub>1</sub>, X<sub>2</sub>,..) is considered. The central idea of this article is to replace the summation over all words of length n b...

When the specialized hardware is not too severe a constraint, the most promising Number Theoretic Transforms are those with 2 as a root of unity, since they can be performed without multiplication. Unfortunately, for a given word length, previously known NTT's with 2 as a root of unity are too short (2^{2^t} +1, 2^{2q}-2^q+1) or too long (32^n + 1)...

We introduce Kloosterman polynomials over F 2 m , and use these polynomials to prove three identities involving Kloosterman sums over F 2 m .

Modulation codes such as runlength-limited codes have been widely employed in magnetic and optical data storage systems. We review the main techniques involved in the design and use of these codes: the maximal code rate or capacity, graphical presentations of constraints, encoders and decoders, and code construction methods such as the ACH state-sp...

The degrees of freedom of reducing simultaneously-switching-outputs limit (SSOL) violations in generating a test pattern set for wiring interconnects were explored without inserting additional test patterns. Algorithmic solutions were presented that took as inputs any interconnect test generation algorithm, a user-defined number of interconnect wir...

We consider lists of distinct q-ary addresses of length n. We wish that any b consecutive addresses in such a list agree in many positions. We give upper bounds on what can be achieved. Moreover, for each q and n, we give explicit constructions of address lists, among which the conventional q-ary reflected Gray code, that attain these bounds for al...

In order to prevent ground bounce, automatic test pattern generation (ATPG) algorithms for wire interconnects have recently been extended with the capability to restrict the maximum Hamming distance between any two consecutive test patterns to a user-defined integer, referred to as simultaneously-switching outputs limit (SSOL). The conventional app...

We give a proof for a conjecture of De Caen and Van Dam [1] concerning the existence of a 4-class association scheme on the set of all unordered pairs of points of the projective line PG(1, q²), where q = 2^m. ), where q = 2 .

We show that a code C of length n over an alphabet Q of size q with minimum distance 2 and covering radius 1 satisfies |C| ≥ qn−1/(n − 1). For the special case n = q = 4 the smallest known example has |C| = 31. We give a construction for such a code C with |C| = 28.

In the literature there exist several methods for errors-anderasures decoding of RS codes. In this paper we present a unified
approach that makes use of behavioral systems theory. We show how different classes of existing algorithms (e.g., syndrome
based or interpolation based, non-iterative, erasure adding or erasure deleting) fit into this framew...

We show the existence of a four-class association scheme defined on the unordered pairs of distinct points from PG(1, q(2)), for q greater than or equal to 4 a power of 2, thereby proving a conjecture of D. de Caen and E. van Darn (Fissioned triangular schemes via the cross-ratio, European J. Combin. 22 (2001), 297-301). This is a fusion of certain...

Using maximal arcs in PG(3, 2m), we give a new proof of the fact that the binary cyclic code C(m)1, 22h−2h+1, the code of length 2m−1 with defining zeroes α and αt, t=22h−2h+1, where α is a primitive element in GF(2m), is 2-error-correcting when gcd(m, h)=1.

Let C t 0 ,t 1 ,⋯,t r denote the binary cyclic code of length n=2 m -1 with defining zeros α t 0 ,α t 1 ,⋯,α t r , where α is a primitive element of GF(2 m ). Using the method in [H. D. L. Hollmann and Q. Xiang, A proof of the Welch and Niho conjectures on cross-correlations of binary m-sequences, Finite Fields Appl. 7, 253-286 (2001; Zbl 1027.9400...

We develop a mathematical framework to investigate classification or ordering of sequential input by means of finite-state algorithms, where the aim is to reduce the "diversity" at the output, that is, to achieve entropy reduction. Our main interest is in optimal time-varying strategies; here, given a (finite) collection of algorithms sharing a com...

Modern packaging technology combined with densely populated assemblies requires efficient design for test features. In particular, modern memories with complex interfaces need to be addressed. This paper presents the details of a test technology that makes assembly test more efficient. The method is based on the implementation of XOR and XNOR gates...

The channel IC described here achieves data rates of 380 Mb/s at
performance levels that improve in various directions upon the state of
the art. It accomplishes these feats in a mature 1 μm CBiCMOS
technology at a readmode power consumption of only 800 mW. The paper
discusses some of the underlying architectural concepts

We show that if the collection of all binary vectors of lengthnis partitioned intokspheres, then eitherk⩽2 ork⩾n+2. Moreover, such partitions withk=n+2 are essentially unique.

Let S be a constrained system of finite type, described in terms
of a labeled graph M of finite type. Furthermore, let C be an
irreducible constrained system of finite type, consisting of the
collection of possible code sequences of some finite-state-encodable,
sliding-block-decodable modulation code for S. It is known that this
code could then be...

Besides timing recovery and automatic gain control, data receivers
often perform adaptive slope or bandwidth control. This note presents a
set of maximum run-length constraints that facilitates the joint
accomplishment of these three tasks. Simple polarity-bit codes that
introduce these constraints are described. The study is of particular
interest...

. An (n, k)-universal sequence is a binary sequence with the property that each window of size k and span at most n is covered by the sequence, i.e., each sequence of length k occurs as the content of a shift of the window. We derive upper and lower bounds on the minimum length of universal sequences,
both for the linear case and the circular case.

Loop structures in software code may reveal essential information about implemented algorithms and their parameters, even if the observer has no knowledge about which instructions are executed. Regular patterns can for instance be observed in power consumption, instruction fetches in external memory, or radiated EM energy. This paper addresses the...

Let S be a constrained system, described in terms of a labelled graph M of finite type. Furthermore, let C be an irreducible constrained system consisting of the collection of possible code sequences of some sliding-block decodable modulation code for S. It is known that this code could then be obtained by state-splitting, using a suitable approxim...

We introduce and investigate the class of bounded-delay-encodable
block-decodable (BDB) codes. Several characterizations for this class of
codes are given, and some construction methods, especially for
one-symbol look-ahead BDB codes, are described. In another direction, we
use our results to show the existence of a decision procedure for some
basi...

We report on block-coding techniques for partial-response channels
with transfer function (1∓D<sup>m</sup>), m=1, 2, ... . We
consider various constructions of block codes with prescribed minimum
Euclidean distance. Upper and lower bounds to the size of a code with
minimum squared Euclidean distance greater than unity are furnished. A
table...

Given a programmable finite-state input/output device, what
program(s) maximally reduce(s) the “diversity” of the
possible output sequences of the device? This question is made precise,
and a method is developed to determine this minimum achievable diversity

We show an algebraic approach for the design of ladder structures
for causal biorthogonal filter banks. The key ingredient of the approach
is known in literature as Euclid's algorithm. Using this algorithm we
derive some strong result on the design freedom for ladder structures.
In particular we show that the dimensionality of the problem plays an...

In this paper we show an algebraic approach for the design of ladder structures for causal bi-orthogonal filter banks. The key ingredient of the approach is known in literature as Euclid's algorithm. Using this algorithm we derive some strong result on the design freedom for ladder structures. In particular we show that the dimensionality of the pr...

Describes a (d,k)=(1,8) runlength-limited (RLL) rate 8/12 code
with fixed codeword length 12. The code is block-decodable; a codeword
can be decoded without knowledge of preceding or succeeding codewords.
The code belongs to the class of bounded delay block-decodable (BDB)
codes with one symbol (8 bits) look-ahead. Due to its format, this code
is p...

We introduce and investigate the class of bounded-delay-encodable block-decodable (BDB) codes. Several characterizations for this class of codes are given, and some construction methods, especially for one-symbol look-ahead BDB codes, are described. In another direction, we use our results to show the existence of a decision procedure for some basi...

We present a technique to construct sliding-block modulation codes with a small decoding window. Our method involves state-splitting and look-ahead coding techniques, and crucially depends on a new, entirely “local”construction method for bounded-delay codes

A code is a collection of words or strings, not necessarily all of
the same length, over come fixed alphabet. A relation is established
between the insertion-and-deletion correcting capability of a code and
its minimum distance for suitable Levenshtein-type distance measures

In digital recorders, the coded information is commonly grouped in
large blocks, called frames. The authors concentrate on the frame
synchronization problem of run-length-limited sequences, or ( d ,
k ) sequences. They commence with a brief description of ( d
, k )-constrained sequences, and proceed with the
examination of the channel capacity. It...

It was claimed [1], [2]that some special digital filters, when used as write equalizers, can be beneficially employed in digital magnetic tape recording to improve the signal-to-noise ratio (SNR). Because of the physical nature of saturation recording, the several digital filters discussed previously are all constrained to transform two-level input...

For a prime p and integer k > 1, let A be an element of the group of nonsingular k x k matrices over the field of integers modulo p. It follows from a theorem of Niven [1] that the order of A in this group is at most p^k - 1. It is conjectured that the order of A is p^k - 1 only when det (A) is a primitive root (mod p). Prove this conjecture or giv...

The problem of appraising the spectral performance of codes based on a new algorithm for generating zero-disparity codewords presented by D.E. Knuth (1986) is addressed. In order to get some insight into the efficiency of Knuth's construction technique, the authors evaluate the spectral properties of its code streams. The structure of Knuth codes a...

The author describes the collection of all possible transfer functions of digital recursive filters that, when operating at p times the original data rate on runlength-limited (RLL) (d,k) bipolar data, can transform each allowable input signal into a bipolar output signal. For all integer values of p, d, and k with p>or=1 and 0<or=d<k<or= infinity...

Several write equalization methods have been proposed for magnetic recording applications with the aim to improve the signal-to-noise ratio (SNR) at the detector. All these write-equalizers may be expressed as digital recursive filters that operate at p times the original data rate, for some positive integer p, and are further constrained to transf...

Consider a shift register (SR) of length n and a
collection of designated subsets of {0,1, . . ., n -1}. The
problem is how to add feedback to the SR such that the resulting linear
feedback shift register (LFSR) exercises (almost) exhaustively each of
the designated subsets and is of small period. Several previously known
results for maximum-length...

The authors have proposed a new type of delay element which can be implemented with simple hardware. Moreover, they have shown that this delay element allows efficient memory sharing if the specified delays to be realised are scheduled for example with the first-fitting algorithm.

A new algorithm is presented for the fast computation of the Discrete Fourier Transform. This algorithm belongs to that class of recently proposed 2<sup>n</sup>-FFT's which present the same arithmetic complexity (the lowest among any previously published one). Moreover, this algorithm has the advantage of being performed "in-place", by repetitive u...

In [5], Mathon described a pseudocyclic symmetric 3-class association scheme on 28 points. In this paper we will first present a general method (a ‘switching’ construction) to construct new symmetric 3-class association schemes from old ones. This method is then applied to the scheme above and yields another scheme with the same parameters. A proof...

First we give a decomposition of an FFT of length 2n into a number of one-dimensional polynomial products. If these products are computed with minimum multiplication algorithms, we show that the 2n FFT can be computed with less than 2n+1 nontrivial complex multiplications. A variation of this algorithm is also shown to give the same multiplication...

In this paper, the Rao-Wilson bound [1], together with the dual of Lloyds Theorem are generalised to t-designs
with repeated points in Q-polynomial association schemes . The proof uses a generalisation of a result of Connor
[5] for classical 2-designs. Moreover, a new proof is given of a sharper version of McWilliams inequality, and
the case of equ...

The most promising Number Theoretic Transforms are those with 2 as a root of unity, since they can be performed without multiplications. One of the main problems is then the complexity of the arithmetic modulo M. We present here a generalized form of the NTT allowing the study of the problems of the NTT's and their arithmetic modulo M together. We...

In this paper, the Rao-Wilson bound [1], together with the dual of Lloyds Theorem are generalised to t-designs with repeated points in Q-polynomial association schemes. The proof uses a generalisation of a result of Connor [5] for classical 2-designs. Moreover, a new proof is given of a sharper version of McWilliams inequality, and the case of equa...

When the specialized hardware is not too severe a constraint, the most promising Number Theoretic Transforms are those with 2 as a root of unity, since they can be performed without multiplication. Unfortunately, for a given wordlength, previously known NTT's with 2 as a root of unity are too short ( 2^{2}^{t} + 1, 2^{2q}-2^{q}+1 ) or too long (3.2...

The letter is an attempt to generalise the advantageous features of Fermat number transforms to match the word length of the modulus of the NTT to the desired dynamic range of the convolution. First, a characterisation of all the transforms with x = 2 and N = 2n is given. We then extend the method to the case N = s Ã 2n.