Pseudo-random sequences and arrays

Binary sequences of length n = 2m- 1 whose autocorrelation function is either 1 or -1/n have been known for a long time, and are called pseudo-random (or PN) sequences, or maximal-length shift-register sequences. Two-dimensional arrays of area n = 2lm- 1 with the same property have rcently been found by several authors. This paper gives a simple description of such sequences and arrays and their many nice properties.

[1]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[2]  E. Wright,et al.  An Introduction to the Theory of Numbers , 1939 .

[3]  E. T. An Introduction to the Theory of Numbers , 1946, Nature.

[4]  de Ng Dick Bruijn A combinatorial problem , 1946 .

[5]  N. Zierler Linear Recurring Sequences , 1959 .

[6]  Irving S. Reed,et al.  Note on the existence of perfect maps , 1962, IRE Trans. Inf. Theory.

[7]  W. W. Peterson,et al.  Error-Correcting Codes. , 1962 .

[8]  Thomas C. Bartee,et al.  Computation with Finite Fields , 1963, Inf. Control..

[9]  Edwin Weiss,et al.  An Acquirable Code , 1964, Inf. Control..

[10]  S. H. Tsao Generation of delayed replicas of maximal-length linear binary sequences , 1964 .

[11]  R. Spann A two-dimensional correlation property of pseudorandom maximal length sequences , 1965 .

[12]  B. Gordon On the existence of perfect maps (Corresp.) , 1966, IEEE Trans. Inf. Theory.

[13]  R. Gold Characteristic Linear Sequences and Their Coset Functions , 1966 .

[14]  J. A. Chang Generation of 5-level maximal-length sequences , 1966 .

[15]  P. D. Roberts,et al.  Statistical properties of smoothed maximal-length linear binary sequences , 1966 .

[16]  K. R. Godfrey Three-level m sequences , 1966 .

[17]  Jean-Marie Goethals Factorization of cyclic codes , 1967, IEEE Trans. Inf. Theory.

[18]  Jack K. Wolf,et al.  On the Synthesis of Two-Dimensional Arrays with Desirable Correlation Properties , 1967, Inf. Control..

[19]  M. Maniere,et al.  Four-level pseudorandom sequences , 1967 .

[20]  J. Savage Some simple self-synchronizing digital data scramblers , 1967 .

[21]  I. G. Cumming Autocorrelation function and spectrum of a filtered, pseudorandom binary sequence , 1967 .

[22]  P. R. Geffe Open letter to communications engineers , 1967 .

[23]  Autocorrelation function of a 4-level msequence , 1968 .

[24]  Neal Zierler,et al.  On Primitive Trinomials (Mod 2) , 1968, Inf. Control..

[25]  E. C. Posner Combinatorial structures in planetary reconnaissance. , 1968 .

[26]  N J Sloane,et al.  Codes for multiplex spectrometry. , 1969, Applied optics.

[27]  H. Fredricksen The lexicographically least de Bruijn cycle , 1970 .

[28]  Elwyn R. Berlekamp Some mathematical properties of a scheme for reducing the bandwidth of motion pictures by hadamard smearing , 1970, Bell Syst. Tech. J..

[29]  Spectrometric imager. , 1971, Applied optics.

[30]  Lih-Jyh Weng Decomposition of M-sequences and its applications , 1971, IEEE Trans. Inf. Theory.

[31]  Akira Fukuda,et al.  A theory of two-dimensional linear recurring arrays , 1972, IEEE Trans. Inf. Theory.

[32]  A. Street,et al.  Combinatorics: room squares, sum-free sets, Hadamard matrices , 1972 .

[33]  U. Henriksson On a Scrambling Property of Feedback Shift Registers , 1972, IEEE Trans. Commun..

[34]  Wayne Stahnke Primitive binary polynomials , 1973 .

[35]  Michael Willett The Index of an M-Sequence , 1973 .

[36]  Robert L. McFarland,et al.  A Family of Difference Sets in Non-cyclic Groups , 1973, J. Comb. Theory A.

[37]  HIDEKI IMAI,et al.  Two-dimensional Fire codes , 1973, IEEE Trans. Inf. Theory.

[38]  E. R. Graf,et al.  The Subsequence Weight Distribution of Summed Maximum Length Digital Sequences , 1974, IEEE Trans. Commun..

[39]  Jacob T. B. Beard,et al.  Computing in GF(q) , 1974 .

[40]  J. T. Harvey High-speed m sequence generation , 1974 .

[41]  Steven A. Tretter Properties of PN2 sequences (Corresp.) , 1974, IEEE Trans. Inf. Theory.

[42]  David E. Carter On the Generation of Pseudonoise Codes , 1974, IEEE Transactions on Aerospace and Electronic Systems.

[43]  Elwyn R. Berlekamp,et al.  Some long cyclic linear binary codes are not so bad , 1974, IEEE Trans. Inf. Theory.

[44]  J. Dillon Elementary Hadamard Difference Sets , 1974 .

[45]  Jacob T. B. Beard,et al.  Some primitive polynomials of the third kind , 1974 .

[46]  D. H. Green,et al.  Irreducible polynomials over composite Galois fields and their applications in coding techniques , 1974 .

[47]  Harold Fredricksen,et al.  A Class of Nonlinear de Bruijn Cycles , 1975, J. Comb. Theory, Ser. A.

[48]  N J Sloane,et al.  Errors in Hadamard spectroscopy or imaging caused by imperfect masks. , 1975, Applied optics.

[49]  Staffan A. Fredricsson Pseudo-randomness properties of binary shift register sequences (Corresp.) , 1975, IEEE Trans. Inf. Theory.

[50]  A. H. Spittle,et al.  High-speed m sequence generation: a further note , 1975 .

[51]  J. Lindner,et al.  Binary sequences up to length 40 with best possible autocorrelation function , 1975 .

[52]  J.L. Smith,et al.  Some cryptographic techniques for machine-to-machine data communications , 1975, Proceedings of the IEEE.

[53]  M. Schroeder Diffuse sound reflection by maximum−length sequences , 1975 .

[54]  David Burton Elementary Number Theory , 1976 .

[55]  O. S. Rothaus,et al.  On "Bent" Functions , 1976, J. Comb. Theory, Ser. A.

[56]  N J Sloane,et al.  Masks for Hadamard transform optics, and weighing designs. , 1976, Applied optics.

[57]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[58]  Underwood Dudley Elementary Number Theory , 1978 .

[59]  B. Poonen LINEAR RECURSIVE SEQUENCES , .