Henk D L Hollmann
Philips Electronics, Philips IP&S, Department Member
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A new algorithm is presented for the fast computation of the Discrete Fourier Transform. This algorithm belongs to that class of recently proposed 2n-FFT's which present the same arithmetic complexity (the lowest among any previously... more
A new algorithm is presented for the fast computation of the Discrete Fourier Transform. This algorithm belongs to that class of recently proposed 2n-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 use of a "butterfly"- type structure, without any data reordering inside the algorithm. Furthermore, it can easily be applied to real and real symmetric data with reduced arithmetic complexity by removing all redundancy in the algorithm.
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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... more
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 $m=2$.
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We give recursive constructions, valid for any field, of [n,k] codes capable of correcting a (wrap-around) burst of n - k erasures.
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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. The... more
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. The raised the question if such [n,k] codes exist for all integers k and n with 1<= k <= n and all fields (in particular, for the binary field). In this note, we answer this question affirmatively by giving two recursive constructions and a direct one.
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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... more
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 approximate eigenvector. In this paper we show that the collection of all approximate eigenvectors that could be used in such a construction of C contains a unique minimal element. Moreover, we show how to construct its linear span from knowledge of M and C only, thus providing a lower bound on the components of such vectors. For illustration we discuss an example showing that sometimes arbitrarily large approximate eigenvectors are required to obtain the best code (in terms of decoding-window size) although a small vector is also available
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We construct a class of permutation polynomials of $\bF_{2^m}$ that are closely related to Dickson polynomials.
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... more
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 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 a and if, ...
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... more
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 Mc Williams dans une version légèrement plus forte, et on traite le cas de l'égalité . Avec e = [t/2], la borne généralisée e de Rao-Wilson devient b >_ t (y) E uj, où b est le nombre total de points, et uo, u L , . . . , u n sont les multiplicités d u j= 0 schéma, si un point y est répété a"( y) fois. Se restreignant aux schémas de Johnson et de Hamming, on trouve (0) , bei pour des t-designs classiques à b blocs, sur v points, si un . bloc i est répété e i fois ([4]), e t b ? ei E (fl)(q_ 1)j, pour des tableaux orthogonaux de force maximale t, à b lignes, de longueur n, sur un alphabe t j=o J à q lettres, si une l...
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ABSTRACT 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... more
ABSTRACT 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 $(i,di+k)$ 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 $\mathrm{DB}(n,p)$.
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ABSTRACT 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... more
ABSTRACT 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 1027.94006)], we determine the weight distribution of the following cyclic codes. (i) C 1,t 1 ,t 2 , where m=2r+1, t 1 =2 r +1, t 2 =2 r-1 +1. (This code appeared in Research Problem 9.7 of F. J. MacWilliams and N. J. A. Sloane [The theory of error-correcting codes, North-Holland (1977; Zbl 0369.94008)]). (ii) C 1,t,t 2 , where m=2r+1, t=1+2 r+1 . (This code appeared in a conjecture of A. Chang, P. Gaal, S. W. Golomb, G. Gong, and P. V. Kumar [On a sequence conjectured to have ideal 2-level auto-correlation function, ISIT 1998, Cambridge, MA (1998)]). (iii) Several cyclic codes in the paper of J. H. van Lint and R. M. Wilson [IEEE Trans. Inf. Theory 32, 23-40 (1986; Zbl 0616.94012)]. (iv) C 1,t , where m=2r, t=∑ i=0 r 2 ik , gcd(m,k)=1.
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ABSTRACT 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... more
ABSTRACT 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 to bypass a functional circuit. It is compatible yet complimentary to boundary-scan. Mathematical proof and simulation results show the effectiveness of this method for detection and diagnosis of assembly faults. Known alternatives are compared for test coverage
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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... more
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
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Let q = pr be a prime power, and let f(x) = xm − �m−1xm−1 − � � � − �1x − �0 be an irreducible polynomial over the finite field GF(q) of size q. A zeroof f is called nonstandard (of degree m) over GF(q) if the recurrence relation um =... more
Let q = pr be a prime power, and let f(x) = xm − �m−1xm−1 − � � � − �1x − �0 be an irreducible polynomial over the finite field GF(q) of size q. A zeroof f is called nonstandard (of degree m) over GF(q) if the recurrence relation um = �m−1um−1 + � � � + �1u1 + �0u0 with characteristic polynomial f can generate the powers ofin a nontrivial way, that is, with u0 = 1 and f(u1) 6= 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 = pr with r ≤ 4. 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 irr...
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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... more
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 approach to meet this SSOL constraint is to insert additional test patterns between consecutive test patterns if their Hamming distance is too large; this approach often leads to many more test patterns than strictly necessary. This paper presents an algorithm that generates, for a user-defined number of interconnect wires, a minimal set of test patterns that respect a user-defined SSOL constraint. Experimental results show that, in comparison to the conventional approach, our algorithm leads to a significant reduction in the test pattern count and corresponding test application time. For example, for problem instances with 5000, 6000, 7000, and 8000 wires, the algorithm reduces the corresponding test application time on average with 45%.
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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... more
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.
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IfCis aq-ary code of lengthnandaandbare two codewords, thencis called a descendant ofaandbifci∈{ai, bi} fori= 1,
, n. We are interested in codesCwith the property that, given any descendantc, one can always identify at least one of the... more
IfCis aq-ary code of lengthnandaandbare two codewords, thencis called a descendant ofaandbifci∈{ai, bi} fori= 1,
, n. We are interested in codesCwith the property that, given any descendantc, one can always identify at least one of the parent codewords inC. We ...
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Abstract: We show the existence of a four-class association scheme defined on the unordered pairs of distinct points from PG(1, q&amp;amp;amp;amp;sup2;), for q &amp;amp;amp;gt;= 4 a power of 2, thereby proving a conjecture of De... more
Abstract: We show the existence of a four-class association scheme defined on the unordered pairs of distinct points from PG(1, q&amp;amp;amp;amp;sup2;), for q &amp;amp;amp;gt;= 4 a power of 2, thereby proving a conjecture of De Caen and Van Dam [2]. This is a fusion of certain relations in the fission scheme FT (q&amp;amp;amp;amp;sup2; + 1) obtained from the triangular association scheme (see [2]). Combining three relations in the above four...
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A general solution of digital recursive filters which produce two-level output is given. To give a rough idea of the performance of write equalizers, a high-density recording channel corrupted by additive noise is appraised, and the... more
A general solution of digital recursive filters which produce two-level output is given. To give a rough idea of the performance of write equalizers, a high-density recording channel corrupted by additive noise is appraised, and the signal-to-noise ratio with and without a write equalizer is compared. Calculations show that a small gain can be expected in using some of the
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ABSTRACT We report on block-coding techniques for partial-response channels with transfer function (1&amp;mnplus;Dm), m=1, 2, ... . We consider various constructions of block codes with prescribed minimum Euclidean distance. Upper and... more
ABSTRACT We report on block-coding techniques for partial-response channels with transfer function (1&amp;mnplus;Dm), 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 is presented of cardinalities of codes of small length with prescribed minimum squared Euclidean distance
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ABSTRACT 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... more
ABSTRACT 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 LFSR are extended to more general LFSR, and in particular a previously known algorithm is simplified and extended. Applications to the problems of VLSI self-testing are discussed and illustrated
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ABSTRACT 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... more
ABSTRACT 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&#39;s construction technique, the authors evaluate the spectral properties of its code streams. The structure of Knuth codes allows the derivation a simple expression for (an approximation to) the sum of variance of these codes. This quantity plays a key role in the spectral performance characterization of DC-balanced codes. The authors evaluate this expression and compare the sum variance of Knuth codes with the sum variance of the polarity bit codes for fixed redundancy. Under the premise that the sum variance can serve as a quantity to judge the width of the spectral notch, the authors conclude that codes based on Knuth&#39;s algorithm offer less spectral suppression than polarity bit codes with the same redundancy.
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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... more
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&amp;amp;amp;amp;amp;amp;amp;amp;gt;or=1 and 0&amp;amp;amp;amp;amp;amp;amp;amp;lt;or=d&amp;amp;amp;amp;amp;amp;amp;amp;lt;k&amp;amp;amp;amp;amp;amp;amp;amp;lt;or= infinity , the collection of all
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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... more
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 in digital magnetic recording
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When testing the interconnect structures on a board, test programmers sometimes ask, How can I control the test pattern generation process to avoid ground bounce problems during Extest mode? Those wishing to satisfy a... more
When testing the interconnect structures on a board, test programmers sometimes ask, How can I control the test pattern generation process to avoid ground bounce problems during Extest mode? Those wishing to satisfy a simultaneously-switching-outputs constraint will find several new solutions in this article.
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ABSTRACT 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... more
ABSTRACT 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, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright Author&amp;apos;s personal copy
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ABSTRACT 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... more
ABSTRACT 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.
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ABSTRACT We introduce Kloosterman polynomials over , and use these polynomials to prove three identities involving Kloosterman sums over .
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Discrete MathentatiO 52 (1904) 209224 209 NorthHolland PSLUDOCYCLIC 3CLASS ASSOCIATION SCH.EMFS ON zg ro* IIenk HO CNETPAB OEJETP, 92131 IssyOMoulineaux, France Received 29 Novembtr 1983 In [5], Mathon describe. O eudocyclic symmetric... more
Discrete MathentatiO 52 (1904) 209224 209 NorthHolland PSLUDOCYCLIC 3CLASS ASSOCIATION SCH.EMFS ON zg ro* IIenk HO CNETPAB OEJETP, 92131 IssyOMoulineaux, France Received 29 Novembtr 1983 In [5], Mathon describe. O eudocyclic symmetric 3class association ...