Handbook of Finite Fields

Table of Contents

Front Cover

HANDBOOK OF FINITE FIELDS

e-ISBN 9781439873823

Contents

Preface

I Introduction

1 History of finite fields
1.1 Finite fields in the 18-th and 19-th centuries
o 1.1.1 Introduction
o 1.1.2 Early anticipations of finite fields
o 1.1.3 Gauss's Disquisitiones Arithmeticae
o 1.1.4 Gauss's Disquisitiones Generales de Congruentiis
o 1.1.5 Galois's Sur la thorie des nombres
o 1.1.6 Serret's Cours d'algbre suprieure
o 1.1.7 Contributions of Schnemann and Dedekind
o 1.1.8 Moore's characterization of abstract finite fields
o 1.1.9 Later developments
2 Introduction to finite fields
2.1 Basic properties of finite fields
o 2.1.1 Basic definitions
o 2.1.2 Fundamental properties of finite fields
o 2.1.3 Extension fields
o 2.1.4 Trace and norm functions
o 2.1.5 Bases
o 2.1.6 Linearized polynomials
o 2.1.7 Miscellaneous results
o 2.1.8 Finite field related books
2.2 Tables
o 2.2.1 Low-weight irreducible and primitive polynomials
o 2.2.2 Low-complexity normal bases
o 2.2.3 Resources and standards

II Theoretical Properties

3 Irreducible polynomials
3.1 Counting irreducible polynomials
o 3.1.1 Prescribed trace or norm
o 3.1.2 Prescribed coefficients over the binary field
o 3.1.3 Self-reciprocal polynomials
o 3.1.4 Compositions of powers
o 3.1.5 Translation invariant polynomials
o 3.1.6 Normal replicators
3.2 Construction of irreducibles
o 3.2.1 Construction by composition
o 3.2.2 Recursive constructions
3.3 Conditions for reducible polynomials
o 3.3.1 Composite polynomials
o 3.3.2 Swan-type theorems
3.4 Weights of irreducible polynomials
o 3.4.1 Basic definitions
o 3.4.2 Existence results
o 3.4.3 Conjectures
3.5 Prescribed coefficients
o 3.5.1 One prescribed coefficient
o 3.5.2 Prescribed trace and norm
o 3.5.3 More prescribed coefficients
o 3.5.4 Further exact expressions
3.6 Multivariate polynomials
o 3.6.1 Counting formulas
o 3.6.2 Asymptotic formulas
o 3.6.3 Results for the vector degree
o 3.6.4 Indecomposable polynomials and irreducible polynomials
o 3.6.5 Algorithms for the gcd of multivariate polynomials
4 Primitive polynomials
4.1 Introduction to primitive polynomials
4.2 Prescribed coefficients
o 4.2.1 Approaches to results on prescribed coefficients
o 4.2.2 Existence theorems for primitive polynomials
o 4.2.3 Existence theorems for primitive normal polynomials
4.3 Weights of primitive polynomials
4.4 Elements of high order
o 4.4.1 Elements of high order from elements of small orders
o 4.4.2 Gao's construction and a generalization
o 4.4.3 Iterative constructions
5 Bases
5.1 Duality theory of bases
o 5.1.1 Dual bases
o 5.1.2 Self-dual bases
o 5.1.3 Weakly self-dual bases
o 5.1.4 Binary bases with small excess
o 5.1.5 Almost weakly self-dual bases
o 5.1.6 Connections to hardware design
5.2 Normal bases
o 5.2.1 Basics on normal bases
o 5.2.2 Self-dual normal bases
o 5.2.3 Primitive normal bases
5.3 Complexity of normal bases
o 5.3.1 Optimal and low complexity normal bases
o 5.3.2 Gauss periods
o 5.3.3 Normal bases from elliptic periods
o 5.3.4 Complexities of dual and self-dual normal bases
o 5.3.5 Fast arithmetic using normal bases
5.4 Completely normal bases
o 5.4.1 The complete normal basis theorem
o 5.4.2 The class of completely basic extensions
o 5.4.3 Cyclotomic modules and complete generators
o 5.4.4 A decomposition theory for complete generators
o 5.4.5 The class of regular extensions
o 5.4.6 Complete generators for regular cyclotomic modules
o 5.4.7 Towards a primitive complete normal basis theorem
6 Exponential and character sums
6.1 Gauss, Jacobi, and Kloosterman sums
o 6.1.1 Properties of Gauss and Jacobi sums of general order
o 6.1.2 Evaluations of Jacobi and Gauss sums of small orders
o 6.1.3 Prime ideal divisors of Gauss and Jacobi sums
o 6.1.4 Kloosterman sums
o 6.1.5 Gauss and Kloosterman sums overfinite rings
6.2 More general exponential and character sums
o 6.2.1 One variable character sums
o 6.2.2 Additive character sums
o 6.2.3 Multiplicative character sums
o 6.2.4 Generic estimates
o 6.2.5 More general types of character sums
6.3 Some applications of character sums
o 6.3.1 Applications of a simple character sum identity
o 6.3.2 Applications of Gauss and Jacobi sums
o 6.3.3 Applications of the Weil bound
o 6.3.4 Applications of Kloosterman sums
o 6.3.5 Incomplete character sums
o 6.3.6 Other character sums
6.4 Sum-product theorems and applications
o 6.4.1 Notation
o 6.4.2 The sum-product estimate and its variants
o 6.4.3 Applications
7 Equations over finite fields
7.1 General forms
o 7.1.1 Affine hypersurfaces
o 7.1.2 Projective hypersurfaces
o 7.1.3 Toric hypersurfaces
o 7.1.4 Artin-Schreier hypersurfaces
o 7.1.5 Kummer hypersurfaces
o 7.1.6 p-Adic estimates
7.2 Quadratic forms
o 7.2.1 Basic definitions
o 7.2.2 Quadratic forms overfinite fields
o 7.2.3 Trace forms
o 7.2.4 Applications
7.3 Diagonal equations
o 7.3.1 Preliminaries
o 7.3.2 Solutions of diagonal equations
o 7.3.3 Generalizations of diagonal equations
o 7.3.4 Waring's problem infinite fields
8 Permutation polynomials
8.1 One variable
o 8.1.1 Introduction
o 8.1.2 Criteria
o 8.1.3 Enumeration and distribution of PPs
o 8.1.4 Constructions of PPs
o 8.1.5 PPs from permutations of multiplicative groups
o 8.1.6 PPs from permutations of additive groups
o 8.1.7 Other types of PPs from the AGW criterion
o 8.1.8 Dickson and reversed Dickson PPs
o 8.1.9 Miscellaneous PPs
8.2 Several variables
8.3 Value sets of polynomials
o 8.3.1 Large value sets
o 8.3.2 Small value sets
o 8.3.3 General polynomials
o 8.3.4 Lower bounds
o 8.3.5 Examples
o 8.3.6 Further value set papers
8.4 Exceptional polynomials
o 8.4.1 Fundamental properties
o 8.4.2 Indecomposable exceptional polynomials
o 8.4.3 Exceptional polynomials and permutation polynomials
o 8.4.4 Miscellany
o 8.4.5 Applications
9 Special functions over finite fields
9.1 Boolean functions
o 9.1.1 Representation of Boolean functions
o 9.1.2 The Walsh transform
o 9.1.3 Parameters of Boolean functions
o 9.1.4 Equivalence of Boolean functions
o 9.1.5 Boolean functions and cryptography
o 9.1.6 Constructions of cryptographic Boolean functions
o 9.1.7 Boolean functions and error correcting codes
o 9.1.8 Boolean functions and sequences
9.2 PN and APN functions
o 9.2.1 Functions from F2n into F2m
o 9.2.2 Perfect Nonlinear (PN) functions
o 9.2.3 Almost Perfect Nonlinear (APN) and Almost Bent (AB)
o 9.2.4 APN permutations
o 9.2.5 Properties of stability
o 9.2.6 Coding theory point of view
o 9.2.7 Quadratic APN functions
o 9.2.8 APN monomials
9.3 Bent and related functions
o 9.3.1 Definitions and examples
o 9.3.2 Basic properties of bent functions
o 9.3.3 Bent functions and other combinatorial objects
o 9.3.4 Fundamental classes of bent functions
o 9.3.5 Boolean monomial and Niho bent functions
o 9.3.6 p-ary bent functions in univariate form
o 9.3.7 Constructions using planar and s-plateaued functions
o 9.3.8 Vectorial bent functions and Kerdock codes
9.4 -polynomials and related algebraic objects
o 9.4.1 Definitions and preliminaries
o 9.4.2 Pre-semi fields, semi fields, and isotopy
o 9.4.3 Semi field constructions
o 9.4.4 Semi fields and nuclei
9.5 Planar functions and commutative semi fields
o 9.5.1 Definitions and preliminaries
o 9.5.2 Constructing affine planes using planar functions
o 9.5.3 Examples, constructions, and equivalence
o 9.5.4 Classi cation results, necessary conditions, and the
o 9.5.5 Planar DO polynomials and commutative semi fields of odd order
9.6 Dickson polynomials
o 9.6.1 Basics
o 9.6.2 Factorization
o 9.6.3 Dickson polynomials of the (k + 1)-th kind
o 9.6.4 Multivariate Dickson polynomials
9.7 Schur's conjecture and exceptional covers
o 9.7.1 Rational function definitions
o 9.7.2 MacCluer's Theorem and Schur's Conjecture
o 9.7.3 Fiber product of covers
o 9.7.4 Combining exceptional covers; the (Fq; Z) exceptional tower
o 9.7.5 Exceptional rational functions; Serre's Open Image Theorem
o 9.7.6 Davenport pairs and Poincar e series
10 Sequences over finite fields
10.1 Finite field transforms
o 10.1.1 Basic definitions and important examples
o 10.1.2 Functions between two groups
o 10.1.3 Discrete Fourier Transform
o 10.1.4 Further topics
10.2 LFSR sequences and maximal period sequences
o 10.2.1 General properties of LFSR sequences
o 10.2.2 Operations with LFSR sequences and characterizations
o 10.2.3 Maximal period sequences
o 10.2.4 Distribution properties of LFSR sequences
o 10.2.5 Applications of LFSR sequences
10.3 Correlation and autocorrelation of sequences
o 10.3.1 Basic definitions
o 10.3.2 Autocorrelation of sequences
o 10.3.3 Sequence families with low correlation
o 10.3.4 Quaternary sequences
o 10.3.5 Other correlation measures
10.4 Linear complexity of sequences and multisequences
o 10.4.1 Linear complexity measures
o 10.4.2 Analysis of the linear complexity
o 10.4.3 Average behavior of the linear complexity
o 10.4.4 Some sequences with large n-th linear complexity
o 10.4.5 Related measures
10.5 Algebraic dynamical systems overfinite fields
o 10.5.1 Introduction
o 10.5.2 Background and main definitions
o 10.5.3 Degree growth
o 10.5.4 Linear independence and other algebraic properties of iterates
o 10.5.5 Multiplicative independence of iterates
o 10.5.6 Trajectory length
o 10.5.7 Irreducibility of iterates
o 10.5.8 Diameter of partial trajectories
11 Algorithms
11.1 Computational techniques
o 11.1.1 Preliminaries
o 11.1.2 Representation of finite fields
o 11.1.3 Modular reduction
o 11.1.4 Addition
o 11.1.5 Multiplication
o 11.1.6 Squaring
o 11.1.7 Exponentiation
o 11.1.8 Inversion
o 11.1.9 Squares and square roots
11.2 Univariate polynomial counting and algorithms
o 11.2.1 Classical counting results
o 11.2.2 Analytic combinatorics approach
o 11.2.3 Some illustrations of polynomial counting
11.3 Algorithms for irreducibility testing and for constructing irreducible polynomials
o 11.3.1 Introduction
o 11.3.2 Early irreducibility tests of univariate polynomials
o 11.3.3 Rabin's irreducibility test
o 11.3.4 Constructing irreducible polynomials: randomized algorithms
o 11.3.5 Ben-Or's algorithm for construction of irreducible polynomials
o 11.3.6 Shoup's algorithm for construction of irreducible polynomials
o 11.3.7 Constructing irreducible polynomials: deterministic algorithms
o 11.3.8 Construction of irreducible polynomials of approximate degree
11.4 Factorization of univariate polynomials
11.5 Factorization of multivariate polynomials
o 11.5.1 Factoring dense multivariate polynomials
o 11.5.2 Factoring sparse multivariate polynomials
o 11.5.3 Factoring straight-line programs and black boxes
11.6 Discrete logarithms overfinite fields
o 11.6.1 Basic definitions
o 11.6.2 Modern computer implementations
o 11.6.3 Historical remarks
o 11.6.4 Basic properties of discrete logarithms
o 11.6.5 Chinese Remainder Theorem reduction:
o 11.6.6 Baby steps-giant steps algorithm
o 11.6.7 Pollard rho and kangaroo methods for discrete logarithms
o 11.6.8 Index calculus algorithms for discrete logarithms infinite fields
o 11.6.9 Smooth integers and smooth polynomials
o 11.6.10 Sparse linear systems of equations
o 11.6.11 Current discrete logarithm records
11.7 Standard models forfinite fields
12 Curves over finite fields
12.1 Introduction to function fields and curves
o 12.1.1 Valuations and places
o 12.1.2 Divisors and Riemann{Roch theorem
o 12.1.3 Extensions of function fields
o 12.1.4 Differentials
o 12.1.5 Function fields and curves
12.2 Elliptic curves
o 12.2.1 Weierstrass equations
o 12.2.2 The group law
o 12.2.3 Isogenies and endomorphisms
o 12.2.4 The number of points in E(Fq)
o 12.2.5 Twists
o 12.2.6 The torsion subgroup and the Tate module
o 12.2.7 The Weil pairing and the Tate pairing
o 12.2.8 The endomorphism ring and automorphism group
o 12.2.9 Ordinary and supersingular elliptic curves
o 12.2.10 The zeta function of an elliptic curve
o 12.2.11 The elliptic curve discrete logarithm problem
12.3 Addition formulas for elliptic curves
o 12.3.1 Curve shapes
o 12.3.2 Addition
o 12.3.3 Coordinate systems
o 12.3.4 Explicit formulas
o 12.3.5 Short Weierstrass curves, large characteristic: y2 = x3 3x + b
o 12.3.6 Short Weierstrass curves, characteristic 2, ordinary case:
o 12.3.7 Montgomery curves: by2 = x3 + ax2 + x
o 12.3.8 Twisted Edwards curves: ax2 + y2 = 1 + dx2y2
12.4 Hyperelliptic curves
o 12.4.1 Hyperelliptic equations
o 12.4.2 The degree zero divisor class group
o 12.4.3 Divisor class arithmetic overfinite fields
o 12.4.4 Endomorphisms and supersingularity
o 12.4.5 Class number computation
o 12.4.6 The Tate-Lichtenbaum pairing
o 12.4.7 The hyperelliptic curve discrete logarithm problem
12.5 Rational points on curves
o 12.5.1 Rational places
o 12.5.2 The Zeta function of a function field
o 12.5.3 Bounds for the number of rational places
o 12.5.4 Maximal function fields
o 12.5.5 Asymptotic bounds
12.6 Towers
o 12.6.1 Introduction to towers
o 12.6.2 Examples of towers
12.7 Zeta functions and L-functions
o 12.7.1 Zeta functions
o 12.7.2 L-functions
o 12.7.3 The case of curves
12.8 p-adic estimates of zeta functions and L-functions
o 12.8.1 Introduction
o 12.8.2 Lower bounds for the rst slope
o 12.8.3 Uniform lower bounds for Newton polygons
o 12.8.4 Variation of Newton polygons in a family
o 12.8.5 The case of curves and abelian varieties
12.9 Computing the number of rational points and zeta functions
o 12.9.1 Point counting: sparse input
o 12.9.2 Point counting: dense input
o 12.9.3 Computing zeta functions: general case
o 12.9.4 Computing zeta functions: curve case
13 Miscellaneous theoretical topics
13.1 Relations between integers and polynomials overfinite fields
o 13.1.1 The density of primes and irreducibles
o 13.1.2 Primes and irreducibles in arithmetic progression
o 13.1.3 Twin primes and irreducibles
o 13.1.4 The generalized Riemann hypothesis
o 13.1.5 The Goldbach problem overfinite fields
o 13.1.6 The Waring problem overfinite fields
13.2 Matrices overfinite fields
o 13.2.1 Matrices of specified rank
o 13.2.2 Matrices of specified order
o 13.2.3 Matrix representations of finite fields
o 13.2.4 Circulant and orthogonal matrices
o 13.2.5 Symmetric and skew-symmetric matrices
o 13.2.6 Hankel and Toeplitz matrices
o 13.2.7 Determinants
13.3 Classical groups overfinite fields
o 13.3.1 Linear groups overfinite fields
o 13.3.2 Symplectic groups overfinite fields
o 13.3.3 Unitary groups overfinite fields
o 13.3.4 Orthogonal groups overfinite fields of characteristic not two
o 13.3.5 Orthogonal groups overfinite fields of characteristic two
13.4 Computational linear algebra overfinite fields
o 13.4.1 Dense matrix multiplication
o 13.4.2 Dense Gaussian elimination and echelon forms
o 13.4.3 Minimal and characteristic polynomial of a dense matrix
o 13.4.4 Blackbox iterative methods
o 13.4.5 Sparse and structured methods
o 13.4.6 Hybrid methods
13.5 Carlitz and Drinfeld modules
o 13.5.1 Quick review
o 13.5.2 Drinfeld modules: definition and analytic theory
o 13.5.3 Drinfeld modules overfinite fields
o 13.5.4 The reduction theory of Drinfeld modules
o 13.5.5 The A-module of rational points
o 13.5.6 The invariants of a Drinfeld module
o 13.5.7 The L-series of a Drinfeld module
o 13.5.8 Special values
o 13.5.9 Measures and symmetries
o 13.5.10 Multizeta
o 13.5.11 Modular theory
o 13.5.12 Transcendency results

III Applications

14 Combinatorial
14.1 Latin squares
o 14.1.1 Prime powers
o 14.1.2 Non-prime powers
o 14.1.3 Frequency squares
o 14.1.4 Hypercubes
o 14.1.5 Connections to affine and projective planes
o 14.1.6 Otherfinite field constructions for MOLS
14.2 Lacunary polynomials overfinite fields
o 14.2.1 Introduction
o 14.2.2 Lacunary polynomials
o 14.2.3 Directions and R edei polynomials
o 14.2.4 Sets of points determining few directions
o 14.2.5 Lacunary polynomials and blocking sets
o 14.2.6 Lacunary polynomials and blocking sets in planes of prime
o 14.2.7 Lacunary polynomials and multiple blocking sets
14.3 affine and projective planes
o 14.3.1 Projective planes
o 14.3.2 affine planes
o 14.3.3 Translation planes and spreads
o 14.3.4 Nest planes
o 14.3.5 Flag-transitive affine planes
o 14.3.6 Subplanes
o 14.3.7 Embedded unitals
o 14.3.8 Maximal arcs
o 14.3.9 Other results
14.4 Projective spaces
o 14.4.1 Projective and affine spaces
o 14.4.2 Collineations, correlations, and coordinate frames
o 14.4.3 Polarities
o 14.4.4 Partitions and cyclic projectivities
o 14.4.5 k-Arcs
o 14.4.6 k-Arcs and linear MDS codes
o 14.4.7 k-Caps
14.5 Block designs
o 14.5.1 Basics
o 14.5.2 Triple systems
o 14.5.3 Difference families and balanced incomplete block designs
o 14.5.4 Nested designs
o 14.5.5 Pairwise balanced designs
o 14.5.6 Group divisible designs
o 14.5.7 t-designs
o 14.5.8 Packing and covering
14.6 Difference sets
o 14.6.1 Basics
o 14.6.2 Difference sets in cyclic groups
o 14.6.3 Difference sets in the additive groups of finite fields
o 14.6.4 Difference sets and Hadamard matrices
o 14.6.5 Further families of difference sets
o 14.6.6 Difference sets and character sums
o 14.6.7 Multipliers
14.7 Other combinatorial structures
o 14.7.1 Association schemes
o 14.7.2 Costas arrays
o 14.7.3 Conference matrices
o 14.7.4 Covering arrays
o 14.7.5 Hall triple systems
o 14.7.6 Ordered designs and perpendicular arrays
o 14.7.7 Perfect hash families
o 14.7.8 Room squares and starters
o 14.7.9 Strongly regular graphs
o 14.7.10 Whist tournaments
14.8 (t; m; s)-nets and (t; s)-sequences
o 14.8.1 (t; m; s)-nets
o 14.8.2 Digital (t; m; s)-nets
o 14.8.3 Constructions of (t; m; s)-nets
o 14.8.4 (t; s)-sequences and (T; s)-sequences
o 14.8.5 Digital (t; s)-sequences and digital (T; s)-sequences
o 14.8.6 Constructions of (t; s)-sequences and (T; s)-sequences
14.9 Applications and weights of multiples of primitive and other polynomials
o 14.9.1 Applications where weights of multiples of a base polynomial
o 14.9.2 Weights of multiples of polynomials
14.10 Ramanujan and expander graphs
o 14.10.1 Graphs, adjacency matrices, and eigenvalues
o 14.10.2 Ramanujan graphs
o 14.10.3 Expander graphs
o 14.10.4 Cayley graphs
o 14.10.5 Explicit constructions of Ramanujan graphs
o 14.10.6 Combinatorial constructions of expanders
o 14.10.7 Zeta functions of graphs
15 Algebraic coding theory
15.1 Basic coding properties and bounds
o 15.1.1 Channel models and error correction
o 15.1.2 Linear codes
o 15.1.3 Cyclic codes
o 15.1.4 A spectral approach to coding
o 15.1.5 Codes and combinatorics
o 15.1.6 Decoding
o 15.1.7 Codes over Z4
o 15.1.8 Conclusion
15.2 Algebraic-geometry codes
o 15.2.1 Classical algebraic-geometry codes
o 15.2.2 Generalized algebraic-geometry codes
o 15.2.3 Functioneld codes
o 15.2.4 Asymptotic bounds
15.3 LDPC and Gallager codes overfinite fields
15.4 Turbo codes overfinite fields
o 15.4.1 Introduction
o 15.4.2 Convolutional codes
o 15.4.3 Permutations and interleavers
o 15.4.4 Encoding and decoding
o 15.4.5 Design of turbo codes
15.5 Raptor codes
o 15.5.1 Tornado codes
o 15.5.2 LT and fountain codes
o 15.5.3 Raptor codes
15.6 Polar codes
o 15.6.1 Space decomposition
o 15.6.2 Vector transformation
o 15.6.3 Decoding
o 15.6.4 Historical notes and other results
16 Cryptography
16.1 Introduction to cryptography
o 16.1.1 Goals of cryptography
o 16.1.2 Symmetric-key cryptography
o 16.1.3 Public-key cryptography
o 16.1.4 Pairing-based cryptography
o 16.1.5 Post-quantum cryptography
16.2 Stream and block ciphers
o 16.2.1 Basic concepts of stream ciphers
o 16.2.2 (Alleged) RC4 algorithm
o 16.2.3 WG stream cipher
o 16.2.4 Basic structures of block ciphers
o 16.2.5 RC6
o 16.2.6 Advanced Encryption Standard (AES) RIJNDAEL
16.3 Multivariate cryptographic systems
o 16.3.1 The basics of multivariate PKCs
o 16.3.2 Main constructions and variations
o 16.3.3 Standard attacks
o 16.3.4 The future
16.4 Elliptic curve cryptographic systems
o 16.4.1 Cryptosystems based on elliptic curve discrete logarithms
o 16.4.2 Pairing based cryptosystems
16.5 Hyperelliptic curve cryptographic systems
o 16.5.1 Cryptosystems based on hyperelliptic curve discrete logarithms
o 16.5.2 Curves of genus 2
o 16.5.3 Curves of genus 3
o 16.5.4 Curves of higher genus
o 16.5.5 Key sizes
o 16.5.6 Special curves
o 16.5.7 Random curves: point counting
o 16.5.8 Pairings in hyperelliptic curves
16.6 Cryptosystems arising from Abelian varieties
o 16.6.1 Definitions
o 16.6.2 Examples
o 16.6.3 Jacobians of curves
o 16.6.4 Restriction of scalars
o 16.6.5 Endomorphisms
o 16.6.6 The characteristic polynomial of an endomorphism
o 16.6.7 Zeta functions
o 16.6.8 Arithmetic on an Abelian variety
o 16.6.9 The group order
o 16.6.10 The discrete logarithm problem
o 16.6.11 Weil descent attack
o 16.6.12 Pairings based cryptosystems
16.7 Binary extension field arithmetic for hardware implementations
o 16.7.1 Preamble and basic terminologies
o 16.7.2 Arithmetic using polynomial operations
o 16.7.3 Arithmetic using matrix operations
o 16.7.4 Arithmetic using normal bases
o 16.7.5 Multiplication using optimal normal bases
o 16.7.6 Additional notes
17 Miscellaneous applications
17.1 Finite fields in biology
o 17.1.1 Polynomial dynamical systems as framework for discrete
o 17.1.2 Polynomial dynamical systems
o 17.1.3 Discrete model types and their translation into PDS
o 17.1.4 Reverse engineering and parameter estimation
o 17.1.5 Software for biologists and computer algebra software
o 17.1.6 Speci c polynomial dynamical systems
17.2 Finite fields in quantum information theory
o 17.2.1 Mutually unbiased bases
o 17.2.2 Positive operator-valued measures
o 17.2.3 Quantum error-correcting codes
o 17.2.4 Period finding
o 17.2.5 Quantum function reconstruction
o 17.2.6 Further connections
17.3 Finite fields in engineering
o 17.3.1 Binary sequences with small aperiodic autocorrelation
o 17.3.2 Sequence sets with small aperiodic autoand crosscorrelation
o 17.3.3 Binary Golay sequence pairs
o 17.3.4 Optical orthogonal codes
o 17.3.5 Sequences with small Hamming correlation
o 17.3.6 Rank distance codes
o 17.3.7 Space-time coding
o 17.3.8 Coding over networks

Bibliography

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