H. Fredricksen, `A Survey of Full Length Nonlinear Shift Register Cycle Algorithms', SIAM Review, Vol. 24, pp. 195--221, 1982  Home/Search   Document Not in Database   Summary   Related Articles   Check

....f(E)S = 0 length(S) Two sequences S t and S 2 are said to be equivalent, S t S2, if one is a cyclic shift of the other. For later reference, we state the following known facts. Fact 1. If S is a sequence whose length is a power of 2, then C(S) c if and only if (E 1) S = 1 length(S) 1] [3]. Fact 2: Let F(n) denote a maximal set of pairwise inequivalent sequences of period 2 [lognl t and complexity n 1. Then, every S F(n) satisfies (E 1)ns = 1 length(s) the cardinality of F(n) is IF(n) 2 [logn] 1, and each of the 2 n binary n tuples appears exactly once in one of the ....

H.M. Fredricksen, "A survey of full length nonlinear shift register cycle algorithms," SIAM Rev., vol. 24, pp. 195-221, Apr. 1982.

....A. De Bruijn Sequences and the Decoding Problem E BRUIJN SEQUENCES, i.e. periodic sequences with elements taken from a finite alphabet in which every possible v tuple of elements appears precisely once in a period (for some v) have been well studied for many years, see, for example, 1] [2]. Many constructions are known, and a useful survey has been given by Fredricksen [2] However, the decoding problem, i.e. the problem. of discovering the position within a particular sequence of any specified v tuple, has been much less well studied. This is notwithstanding the fact that for ....

.... sequences with elements taken from a finite alphabet in which every possible v tuple of elements appears precisely once in a period (for some v) have been well studied for many years, see, for example, 1] 2] Many constructions are known, and a useful survey has been given by Fredricksen [2]. However, the decoding problem, i.e. the problem. of discovering the position within a particular sequence of any specified v tuple, has been much less well studied. This is notwithstanding the fact that for certain well known practical applications of de Bruijn sequences, including their use ....

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H. Fredricksen, "A survey of full length nonlinear shift register cycle algorithms," SIAM Rev., vol. 24, pp. 195-221, 1982.

....Diagram (STD) of the DSR. The STD consists of 2 states where d is the length of the DSR. Any state has exactly two possible next states, corresponding to shifting right and inserting the bit 0 or 1 . This STD is referred to commonly in the relevant literature as the DeBruijn diagram [5]. Constructing the STD enables the calculation of the minimal number of bits needed to traverse from one state to the other. The minimal number of bits needed to go from one state to the other is stored in a distance matrix with each entry indicating the minimum number of shifts needed to reach ....

H. Fredricksen. A survey of full length nonlinear shift register cycle algorithms. SIAM Review, 24(2):195-- 221, 1982.

....of functions on nine variables. For a complete description of this figure, the reader is referred to Section 2. Fig. 1a represents the AND function, while Fig. 1b represents the deBruijn function, so called because it corresponds to a Hamiltonian cycle in the deBruijn graph B 3 (Fredricksen [5] and Wegener [10] A deBruijn function is a function on n = 2 k 2 variables, represented by a string s that contains one (and only one) copy of every possible substring of length k. Nodes in this BDD correspond to every distinct substring of s. The number of such nodes is asymptotic to ....

H. Fredricksen, "A Survey of Full Length Nonlinear Shift Register Cycle Algorithms, " SIAM Review, vol. 24, no. 2, pp. 195-220, Feb. 1982.

....v is cyclic sequence of length n v such that every possible v tuples occurs precisely once in a period of the de Bruijn sequence, as a contiguous part. For example, a 2 ary de Bruijn sequence of span 2 is 010. A problem that can be solved with Ouroborean Rings is the Baltimore Hilton Inn problem[3]. A cipher lock system uses a 4 digit code to allow access. Assuming that the code consists of digits from 0 to 9, there are 10 possible codes for the lock. An attacker may try every possible combination, requiring 4 keypresses, or 2 on average. But since the lock opens whenever the ....

....or 2 on average. But since the lock opens whenever the correct code is the last 4 digits, fewer keypresses are required. The best sequence of attack is the Ouroborean ring associated with the lock. Length ) These sequences are called full length nonlinear shift register cycles by Fredricksen[3]. The first solution was offered by Flye Sainte Marie in the last century[10] An important application of the Ouroborean rings is the generation of pseudo random binary sequences of maximal length[6] Other applications have a long history[14] 3] 6 Results 6.1 Length of Optimal Solutions The ....

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H. Fredricksen, A survey of full length nonlinear shift register cycle algorithm, SIAM Rev., 24 (1982), 195-221.

....is a cyclic k ary sequence of length R with the property that every k ary n tuple appears exactly once contiguously on the cycle (of course, R = k n ) First invented in 1894 by Flye St. Marie [6] they were rediscovered in 1946 by de Bruijn [1] and Good [8] An excellent survey by Fredericksen [7] introduces the reader to a vast literature on the topic. An (R; S; m;n) k de Bruijn torus is a toroidal k ary R S array with the property that every k ary m n matrix appears exactly once contiguously on the torus (of course, RS = k mn ) The simplest example of such a torus is the (4; 4; ....

H.M. Fredricksen, A survey of full length nonlinear shift register cycle algorithms, SIAM Review 24 (1982), 195-221.

....1. The binary de Bruijn graph order 3. B k is the line graph of B k 1 , and the latter is easily seen to be Eulerian, so the former must be Hamiltonian. By de Bruijn s theorem, the number of Hamiltonian cycles is 2 2 k 1 k , and there is an elegant algorithm to enumerate these cycles, see [5, 4]. The algorithm uses suitable preference functions f : 2 k 2 that determine which edge to choose when a node u 2 2 k is encountered for the rst time. The opposite edge is chosen at the second encounter. By attening a Hamiltonian cycle into a bit sequence of length 2 k k 1 we obtain a ....

H. Fredricksen. A survey of full length nonlinear shift register cycle algorithms. SIAM Review, 24(2):195{ 221, 1982.

....for Higher Dimensional Perfect Multifactors Garth Isaak Abstract We extend a construction technique for perfect multifactors sometimes called the inverse of Lempel s Homomorphism to dimensions three and higher. 1 Introduction What has come to be known as a de Bruijn cycle (see [4] for more history) is a periodic k ary string in which every k ary substring of a given size appears exactly once (periodically) For example, in the period nine string 001121022j001121022j0011 : each ternary string of length two appears exactly once with period nine. We will usually ....

H.M. Fredricksen, A survey of full length nonlinear shift register cycle algorithms, SIAM Review 24 (1982), 195--221. 17

....i j = h is the d 1 dimensional torus consisting of all entries a I for which i j = h. So, for example, in 2 dimensions the projection along i 2 = 7 is the 7 th column (after the 0 th column) A 1 dimensional De Bruijn torus is what has come to be known as a De Bruijn cycle. See Frederickson [3] for a survey of De Bruijn cycles. Two dimensional De Bruijn tori are examined in Cock [1] Fan et al. 2] and others. See Hurlbert and Isaak [4] for references. A 2 dimensional De Bruijn torus (r 1 ; r 2 ; n 1 ; n 2 ) 2 k is square if n 1 = n 2 and r 1 = r 2 . Except for small values of n j , ....

H.M. Fredricksen, A survey of full length nonlinear shift register cycle algorithms, SIAM Rev., 24 (1982), 195-221.

....(n,m) de Bruijn sequence) is a string of n m symbols 12 . m n ssssuch that each substring of length m, 12 . iiim sss , 1) is unique with subscripts in (1) taken modulo n m . For m2 and n2, there are N= 1 1 (1) m m n nm nn (n,m) de Bruijn sequences (Fredricksen [3]) For example, there are 36 pairs (i.e. m=2) which may be formed using n=6 symbols. The number of (6,2) de Bruijn sequences is over 3.8710 15 . During the 70s 80s, de Bruijn sequences were well studied and several algorithms have been proposed for generating such sequences, e.g. Fredricksen ....

.... use concepts of either finite field theory or combinatorial theory to generate a single (n,m) de Bruijn sequence, but in addition, there is a well known algorithm to sample, with equal probability, random (n,m) de Bruijn sequences [1] An excellent survey has been provided by Fredricksen [3]. Owing to the special properties of de Bruijn sequences, there are various recent applications of (n,m) de Bruijn sequences, such as the planning of reaction time experiments (Emerson and Tobias [2] and Sohn et al. 11] and 3 D pattern recognition (Griffin et al. 6] Hsieh [7 9] and Yee and ....

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H. Fredricksen, A survey of full length nonlinear shift register cycle algorithms, SIAM Review 24 (1982), 195-221.

....to give an upper bound of the number of units. Note that however, its inexistence is not proven if some range in the GC content is allowed. When an arbitrary GC content is allowed, the circular sequence is called a De Bruijn sequence, and the algorithm for its generation is studied extensively [3, 8]. Table 1: Table of n and corresponding Nn n 2 4 6 8 Nn 8 96 1280 26880 2.5 Summary of Constraints In summary, the constraints to be considered in sequence design are as follows. Constraints from Problem Design Restriction sites should appear only at planned positions. Constraints for ....

Fredricksen, H. \A Survey of Full Length Nonlinear Shift Register Cycle Algorithms", SIAM Review 24(2):195-221, 1982.

....is discussed first. Then a coding for some special cases of a cycle plus chords is discussed. 11 4. 1 Relevant Results on Cycles in DM(k) Coding of the vertices of a cycle of length 2 k with a shift register coding has been the subject of considerable interest for coding theory researchers, [8, 9, 14, 15, 25, 26] to name a few. However, in this paper our interest is not just in cycles of length 2 k but also in cycles of an arbitrary length and the only previous work on that subject is by Yoeli[26] The key result for our purpose in Yoeli s paper is that DM(k) has a cycle of every length from 1 to 2 k ....

H. Fredricksen. A survey of full length nonlinear shift register cycle algorithms. SIAM Rev., 24:195--220, April 1982.

.... that a de Bruijn sequence of order n corresponds to an Euler tour in the de Bruijn digraph whose vertices are the binary n tuples, with one edge (x 1 : x n Gamma1 ) x 2 : x n ) for every binary n tuple x 1 : x n [dB46, Mar34] This result has been generalized for k ary n tuples [Fre82], for higher dimensions (de Bruijn tori) Coc88, FFMS85] and for k ary tori [HI95] It is known also, for any m satisfying n m 2 n , that there is a cyclic binary sequence of length m in which no n tuple appears more than once [Yoe62] So, the de Bruijn graph contains cycles of all lengths ....

H. Fredricksen. A survey of full length nonlinear shift register cycle algorithms. SIAM Review, 24(2):195--221, 1982. 34

....survey article of Allouche [A94] The proof of our results centers on a detailed analysis of a version of the de Bruijn graph which appeared first implicitly in [F94] and explicitly in [R83] Good [G46] and de Bruijn [B46] independently defined a version of these graphs in 1946. See Fredricksen [F82] for more references for the de Bruijn graph. Observe that f(1) F 3 = 2, which means the number of distinct subwords of length 1 is 2. Thus we need only consider binary words over 0, 1 in the rest of this paper. We divide the presentation of the proof into 4 parts: 1. Existence 2. Structure ....

Fredricksen, H. A survey of full length nonlinear shift register cycle algorithms, SIAM Rev. 24 (1982), 195-221.

.... the memory strand be composed of only pyrimidines (or purines) and the stickers of only purines (or pyrimidines) Mir] The applied mathematics literature on comma free codes and on de Bruijn sequences (when D = 1) contains detailed discussions of many of the important issues (see [Neveln] and [Fredricksen] for reasonable introductions) Also, Smith, Baum] have discussed sequence design in the context of DNA computation. Finally, D 1 would be reduced if higher affinity PNA or DNG stickers were used; furthermore, D 3 would possibly be reduced to zero. Other variables other than or in addition to ....

Harold Fredricksen. A Survey of Full Length Nonlinear Shift Register Cycle Algorithms. SIAM Review, 24(2):195-221, 1982.

....on sequences with period 2 n for which an entire period of the sequence is known. While this appears to be an algorithm with limited general applicability the mathematical foundations were used to prove some very interesting results [40, 38] in the study of what are called de Bruijn sequences [15, 35]. Very recently the validity of this algorithm has been extended by Blackburn [9] to sequences with any period, although an entire period of the sequence is still required for its application. 4.2.2 Other measures of complexity As a generalization of the linear complexity a great deal of research ....

H. Fredricksen. A survey of full length nonlinear shift register cycle algorithms. SIAM Journal on Applied Mathematics, 24(2):195--221, 1982. REFERENCES 33

.... that they admit simple routing strategies, suggest that they might be useful for the solution of certain interconnection problems arising in communication networks and multiprocessor systems [15] Moreover, de Bruijn Good graphs are of interest in other research areas, for instance in cryptography [6, 10, 14, 24] where one of the problems considered is the generation of random looking pseudo random sequences to be used as the keystream in so called additive stream ciphers. One popular way of generating these sequences is by the use of nonlinear feedback shift registers whose state transition diagram is ....

....will not be considered in this paper. For the special case of binary de Bruijn Good graphs (i.e. q = 2) previous authors have treated the problems of counting the number of Hamiltonian cycles of maximal length 2 n [7] generating these maximal length cycles (called de Bruijn sequences) [10], counting the number fi (2) n; k) of cycles of a certain length k [2, 3, 23] and determining the maximal number of disjoint cycles into which G (2) n can be decomposed [12, 17, 21] The binary case is of special interest because it is best suited for implementations in digital electronics. ....

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H. Fredricksen, A survey of full length nonlinear shift register cycle algorithms, SIAM Review, Vol. 24, No. 2, pp. 195-221, 1982.

....survey article of Allouche [A94] The proof of our results centers on a detailed analysis of a version of the de Bruijn graph which appeared first implicitly in [F94] and explicitly in [R83] Good [G46] and de Bruijn [B46] independently defined a version of these graphs in 1946. See Fredricksen [F82] for more references for the de Bruijn graph. Observe that f(1) F 3 = 2, which means the number of distinct subwords of length 1 is 2. Thus we need only consider binary words over f0; 1g in the rest of this paper. We divide the presentation of the proof into 4 parts: 1. Existence 2. Structure of ....

Fredricksen, H. A survey of full length nonlinear shift register cycle algorithms, SIAM Rev. 24 (1982), 195-221.

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H. Fredricksen, A survey of full length nonlinear shift register cycle algorithms, SIAM Review 24 (2) (1982) 195--221.

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H. Fredricksen, `A Survey of Full Length Nonlinear Shift Register Cycle Algorithms', SIAM Review, Vol. 24, pp. 195--221, 1982

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Harold Fredricksen, A survey of full length nonlinear shift register cycle algorithms, SIAM Review 24 (1982), no. 2, 195--221.

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H. Fredricksen, A survey of full length nonlinear shift register cycle algorithm, SIAM Rev., 24 (1982), 195-221.

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Fredricksen, H.: A survey of full length nonlinear shift register cycle algorithms. SIAM Rev. 24 (1982) 195--221

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H. Fredricksen, A survey of full length nonlinear shift register cycle algorithms, Society of Industrial and Applied Mathematics Review 24 (2) (1982) 195--221.

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H. Fredricksen, \A Survey of Full Length Nonlinear Shift Register Cycle Algorithms," SIAM Review, Vol. 24, pp. 195-221, 1982.

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H. Fredricksen, A survey of full length nonlinear shift register cycle algorithms, SIAM Review 24 (1982), 195--221.

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H. Fredricksen, "A Survey of full length nonlinear shift register cycle algorithms SIAM Review Vol 24, & No.2, pp 195-221, April 1982

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H. Fredricksen, A survey of full length nonlinear shift register cycle algorithms, SIAM Review 24 (1982), 195-221. 12

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