Henk D L Hollmann
Philips Electronics, Philips IP&S, Department Member
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ABSTRACT
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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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