List of All Methods

Linear Constructions

NameUsage Count
Total:58,236
OA with Only One Run / Dual of Code Without Redundancy16
OOA with Only One Run / Dual of NRT-Code Without Redundancy40
Arbitrary OA / Dual of Arbitrary Code2,187
Arbitrary Net2,203
Arbitrary OOA / Dual of Arbitrary NRT-Code6,078
Arbitrary OA / Dual of Arbitrary Code (with Infinite s)16
Arbitrary Net (with Infinite s)16
Arbitrary OOA / Dual of Arbitrary NRT-Code (with Infinite s)40
Reed–Solomon Code464
Reed–Solomon Codes for OOAs1,564
Extended Reed–Solomon Code448
Extended Reed–Solomon Codes for OOAs1,581
Complete OA / Dual of Code with Only One Code Word2,187
Complete OOA / Dual of NRT-Code with Only One Code Word2,124
Repetition Code2,203
Repetition NRT-Code2,092
Repetition Code with Arbitrary Length16
Repetition NRT-Code with Arbitrary Length40
OOA with Large Number of Runs16
Large High-Dimensional Nets with Fixed Strength2,187
Explicit Generator Matrices for OOAs by Edel5
OOAs Constructed from BCH Codes45
Parity-Check Code16
Hamming Code2,171
Simplex Code28
Reed–Muller Code53
Extended Quadratic Residue Code42
Codes of Belov Type457
Caps in Base b = 2258
Oval8
Hyperoval7
Ovoid15
Large Caps in PG(4, b)13
Large Caps in PG(5, b)14
Large Caps in PG(u, 3)7
Product of Two Caps with Tangent Hyperplane32
Caps Completed Using a Computer Search19
Codes from Dual Arcs29
Union of Pairwise Disjoint Baer Subplanes13
Hermitian Unital3
Projective Codes Based on the Barlotti Arcs6
Denniston Codes8
Projective Code from Ovoid6
Codes by Danev and Olsson14
Codes by De Boer and Brouwer56
Augmented Codes by De Boer and Brouwer12
BCH-Codes4,545
Primitive BCH-Codes7,390
Extended Narrow-Sense BCH-Codes1,617
Primitive Extended Narrow-Sense BCH-Codes2,501
Primitive Narrow-Sense BCH-Codes (Sporadic Result)3
Primitive Expurgated Narrow-Sense BCH-Codes (Sporadic Result)3
Narrow-Sense BCH-Codes1,337
Primitive Narrow-Sense BCH-Codes638
Expurgated Narrow-Sense BCH-Codes496
Primitive Expurgated Narrow-Sense BCH-Codes980
Narrow-Sense BCH-Codes (BCH-Bound)255
Expurgated Narrow-Sense BCH-Codes (BCH-Bound)2,374
Cyclic Codes (BCH-Bound)140
Primitive Cyclic Codes (BCH-Bound)262
Narrow-Sense BCH-Codes (Simple Roos-Bound)167
Narrow-Sense BCH-Codes (General Roos-Bound)175
Expurgated Narrow-Sense BCH-Codes (General Roos-Bound)7
Cyclic Codes (General Roos-Bound)1
Primitive Cyclic Codes (General Roos-Bound)5
Generator Matrices Provided on Edel’s Homepage29
Quaternary Constacyclic Codes with Minimum Distance 521
Ternary Negacyclic Codes with Minimum Distance 58
Extended or Lengthened BCH Codes42
Extended or Lengthened Subfield Codes of Reed–Solomon Codes9
Codes Obtained from BCH Codes, Concatenation and Construction X35
Linear Codes Obtained from Additive Codes and Concatenation25
BCH Codes Lengthened with UEP Codes33
Codes Constructed by Inverting Construction Y143
Extended or Lengthened BCH Codes (Additions)26
BCH Codes Lengthened with UEP Codes (Additions)2
Code Obtained from BCH Codes with Construction X42
Quasi-Cyclic and Quasi-Twisted Codes by Boukliev3
Quasi-Cyclic Codes by Boukliev, Daskalov, and Kapralov2
Quasi-Cyclic Codes by Gulliver and Bhargava66
Quasi-Cyclic Codes by Bierbrauer and Gulliver23
Quaternary Quasi-Twisted Codes of Dimension 6 by Gulliver and Östergård18
Quasi-Cyclic and Quasi-Twisted Codes by Daskalov and Gulliver135
Cyclic, Quasi-Cyclic, and Quasi-Twisted Codes by Daskalov, Hristov, and Metodieva165
Codes by Kohnert and Zwanzger35
Golay Code2
Extended Golay Code2
Tetracode1
Hexacode1
Hill Cap1
Glynn Cap1
A Cap Found by Tallini1
A Cyclic Code with Given Generator5
Linear Codes with Explicit Generator Matrix62
A Code by Marcugini, Milani, and Pambianco2
A Code by Bierbrauer, Marcugini, and Pambianco3
k = 2 Construction2,171
Nets Constructed from Net-Embeddable BCH Codes123
Sobol Sequence (Bratley–Fox Implementation) with Equidistant Coordinate4
Niederreiter Sequence (Bratley–Fox–Niederreiter Implementation)10
Niederreiter Sequence (Bratley–Fox–Niederreiter Implementation) with Equidistant Coordinate211
Niederreiter–Xing Sequence (Piršić Implementation)3
Niederreiter–Xing Sequence (Piršić Implementation) with Equidistant Coordinate245
Shift-Nets135
Hand-Crafted Matrix by Schmid and Wolf2
Linear Code Embeddings Found in a Computer Search129
Salzburg Tables30
Van der Corput Sequence35
Faure Sequence4
Generalized Faure Sequence31
Sobol Sequence2
Niederreiter Sequence222
Arbitrary (t, s)-Sequence for Large s32
Niederreiter–Xing Sequence Construction III with García–Stichtenoth Tower As Constant Field Extension106
Construction for Strength k = 12,187

Non-Linear Constructions

NameUsage Count
Total:24
Nordstrom–Robinson OA1
Kerdock Codes10
Delsarte–Goethals Codes10
Preparata Codes3

Linear Existence Results

NameUsage Count
Total:245,136
Gilbert–Varšamov Bound for OAs49,446
Gilbert–Varšamov Bound for Nets42,327
Gilbert–Varšamov Bound for OOAs152,285
Code from Brouwer’s Database1,070
Code from Brouwer’s Database2
Plane (s,3)-Arcs in PG(2,b)2
Existence of Nets from Net-Embeddable BCH Codes4

Bounds (for the Linear and Non-Linear Case)

NameUsage Count
Total:402,499
Trivial Upper Bound on the Number of Factors8,812
Trivial Upper Bound on s6,593
Trivial Upper Bound on the Number of Factors for OOAs32,883
Singleton Bound2,251
Trivial Lower Bound on t2,219
Singleton Bound for OOAs21,846
Trivial Lower Bound on t for Sequences32
Lower Bound on the Number of Runs for OAs with Strength k = 2 and Arbitrarily Large Number of Factors s32
Lower Bound on t for Nets with Strength k = 2 and Arbitrarily Large Dimension s16
Lower Bound on the Number of Runs for OOAs with Strength k = 2 and Arbitrarily Large Number of Factors s80
Rao or (Dual) Hamming Bound52,374
Rao or (Dual) Hamming Bound (Large M)2,062
Generalized Rao Bound for Nets53,905
Generalized Rao Bound for OOAs196,401
(Dual) Plotkin Bound1,061
Generalized (Dual) Plotkin Bound for OOAs4,338
(Dual) Plotkin Bound for n = 24,438
Generalized (Dual) Plotkin Bound for n = 2130
LP Bound with Quadratic Polynomials for OOAs103
Bound Derived from the LP Bound by Trinker24
Linear Programming Bound5,291
Linear Programming Bound for OOAs782
Bound for OAs with Strength k = 22,171
Bose–Bush Bound for OAs with Strength 369
Bound for OAs with Index Unity14
Bound by Seiden and Zemach1
Bound by Seiden1
Trivial Upper Bound on m2,399
Mutually Orthogonal Hypercube Bound2,171

Bounds Applicable Only to the Linear Case

NameUsage Count
Total:21,971
Trivial Bound on the Size of Affine Caps2,095
Affine Subcap of Ovoid Is Optimal15
Sporadic Bounds on the Size of Affine Caps3
Griesmer Bound3,023
Johnson Bound14,244
Improved Johnson Bound2,170
Linear Programming Bound with Additional Constraints on the Weights19
Bound for MDS Codes80
Bound for Almost-MDS-Codes5
Bound for Almost-MDS-Codes with Dimension n = 312
Sharpened Johnson Bound16
Improvement by Maruta on the Griesmer Bound10
Ovoid Cap Is Optimal15
Bound for Caps in PG(4, b)9
20 Is the Largest Size of a Cap in PG(4, 3)1
41 Is the Largest Size of a Cap in PG(4, 4)1
A Result by Edel, Storme, and Sziklai2
A Result by Gronchi and Heim1
A Result by Barát, Edel, Hill, and Storme1
A Result by Landjev, Maruta, and Hill1
A Result by Dodunekov and Landjev2
A Result by Hill and Landjev2
A Result by Hill1
A Result by Boukliev, Kapralov, Maruta, and Fukui4
Bound by Simonis1
Bound by Fontaine and Peterson1
Bound on Codes from Brouwer’s Database236
Bound on Codes from Brouwer’s Database0
A Bound by Schmid and Wolf1

Algebraic Function Fields

NameUsage Count
Total:979
Rational Function Field35
Hermitian Function Field7
Suzuki Function Field2
Klein Quartic1
An Explicitly Constructive Algebraic Function Field88
Function Fields by García and Quoos26
Function Fields by Sémirat48
Sharp Bound on the Number of Rational Points on Curves with Genus 16
Sharp Bound on the Number of Rational Points on Curves with Genus 29
Fields by Serre with Genus 3 or 42
Drinfeld Modules of Rank 1245
Algebraic Function Fields over ℤ3 by Niederreiter/Xing24
Algebraic Function Fields over ℤ5 by Niederreiter/Xing51
Algebraic Function Fields over F9 and F27 by Niederreiter/Xing1
Algebraic Function Fields by Auer11
Algebraic Function Fields by Keller3
Algebraic Function Fields by Beelen and Pellikaan5
Algebraic Function Fields by Teo19
Algebraic Function Fields over ℤ7 by Yeo5
List of Curves by van der Geer and van der Vlugt311
Table by Niederreiter and Xing7
Tower of Function Fields by García and Stichtenoth9
Second Tower of Function Fields by García and Stichtenoth16
Tower of Function Fields by Niederreiter and Xing Based on the Tower by García and Stichtenoth8
Tower of Function Fields by García, Stichtenoth, and Rück13
Tower of Function Fields by Bezerra and García17
Tower of Function Fields over F8 by van der Geer and van der Vlugt5
Tower of Function Fields by Bezerra, García, and Stichtenoth5

Propagation Rules

NameUsage Count
Total:2,585,003
Overall Total:3,313,848
Strength Reduction52,759
t-Expansion116,192
Strength Reduction for OOAs241,253
t-Expansion for Sequences3,453
Code Embedding in Larger Space44,827
NRT-Code Embedding in Larger Space56,041
Duplication / Taking a Subcode of the Dual Code2,171
Net Duplication27,956
OOA Duplication142,960
Truncation182,896
Truncation for OOAs289,350
m-Reduction176,691
m-Reduction from Arbitrarily Large Net2,406
m-Reduction for OOAs81,267
Repeating Each Code Word5,184
Repeating Each Code Word for NRT-Codes12,868
Trace Code5,602
Trace Code for Nets19,996
Trace Code for OOAs4,956
Base Expansion for Nets6
Base Expansion for OOAs45
Base Expansion for Sequences10
Base Reduction0
Base Reduction for Sequences306
Base Reduction for Projective Spaces28
Base Reduction for Projective Spaces (for Nets)322
Discarding Parts of the Base for OAs8,347
Discarding Parts of the Base for OOAs2,064
Direct Product of Two Codes0
Direct Product of Two Nets0
Direct Product of Two OOAs0
(uu+v)-Construction12,144
(uu+v)-Construction for Nets26,842
(uu+v)-Construction for OOAs44,177
Juxtaposition2,852
Juxtaposition for OOAs9,217
Concatenation of Two Codes11,882
Concatenation of Two NRT-Codes11,658
Concatenation of Two Orthogonal Arrays0
Concatenation of Two OOAs0
Construction X with Reed–Solomon Codes1,337
Construction X with Algebraic-Geometric Codes17,795
Construction X with Algebraic-Geometric NRT-Codes67,933
Construction XX with a Chain of Algebraic-Geometric Codes312
Construction X with OOAs from (ts)-Sequence511
Depth Reduction170,049
Extracting Embedded Orthogonal Array5,262
Extracting Embedded OOA46,791
Appending Arbitrary Columns7,012
Net Defined by OOA35,533
Embedding of OOA with Gilbert–Varšamov Bound81,977
Niederreiter–Xing Sequence Construction II/III513
Niederreiter–Xing Sequence Construction III Using Non-Rational Places26
Extended Algebraic-Geometric Codes8,663
Extended Algebraic-Geometric NRT-Codes63,666
Algebraic-Geometric Codes4,531
Algebraic-Geometric NRT-Codes13,927
Algebraic-Geometric Codes with Known Gap Numbers19
Algebraic-Geometric NRT-Codes with Known Gap Numbers32
Algebraic-Geometric Codes Defined Using a Non-Rational Place399
Algebraic-Geometric NRT-Codes Defined Using a Non-Rational Place617
A Recursive Bound on the Size of Affine Caps by Bierbrauer and Edel53
Base Reduction for Affine Caps3
Every Affine Cap Is Also a Projective Cap18
Doubling a Cap (and Obtaining an Affine Cap)11
Adding a Parity Check Bit1,208
Residual Code1,828
Dual of MDS Code Is Again an MDS Code14
Dual of a Near-MDS Code Is Again a Near-MDS Code13
Contraction23
Contraction (with Narrow-Sense BCH-Code)57
Contraction (with Expurgated Narrow-Sense BCH-Code)63
(uuvu+v+w)-Construction2,311
Generalized (uu+v)-Construction1,022
Construction Y1 (Bound)536
Construction Y168
Dual Code (with Bound on d by Construction Y1)6
Varšamov–Edel Lengthening12,524
Removing Affine Subspaces38
Hill Recurrence5
Doubling a Cap10
Product of Projective Cap with (Hyper)oval9
Product of Projective Cap with b2-Ovoid Cap in AG(3, b)7
Product of Projective Cap with Cap in AG(4, b)9
Product of Projective Cap with Cap in AG(5, b)6
Product of Projective Cap with Cap in AG(6, b)9
Product of Two Projective Caps and 2-Cap in AG(1, b)1
Product of Two Projective Caps and (Hyper)oval2
Product of Two Projective Caps and an Affine Cap Avoiding Two Hyperplanes2
Construction for [ss−6, 6]-Code from [s, 3, s−3]-Code4
Construction X with Cyclic Codes9,004
Construction XX with Cyclic Codes16,740
Construction XX with a Chain of Cyclic Codes788
Construction X4 with Cyclic Codes155
Construction X with Extended Narrow-Sense BCH Codes17,089
Construction XX with a Chain of Extended Narrow-Sense BCH Codes3,895
Construction X4 with Extended Narrow-Sense BCH Codes424
Construction X with De Boer–Brouwer Codes201
Construction XX with De Boer–Brouwer Codes9
Construction XX with a Chain of De Boer–Brouwer Codes181
Construction X with Varšamov Bound4,202
Construction X with Varšamov–Edel Bound13
Construction X with Ovoid11
Construction X with Kerdock and Delsarte–Goethals OAs19
Construction XX with a Chain of Delsarte–Goethals OAs0
Construction X4 with Kerdock OAs17
Standard Lengthening for NRT-Codes32
Net from Sequence81,720
OOA from Sequence284
Logical Equivalence (for Sequences)9,548
Logical Equivalence (for Nets with Unbounded m)9,548
Base Change2,828
Appending kth Column1,281
OAs with Strength 3, b ≠ 2, and m > 3 Are Always Embeddable123
OOA Folding238,893
OA Folding and Stacking4,896
OOA Folding and Stacking with Additional Row26,942
OOA Stacking with Additional Row10,667

Other Descriptions