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List of All Methods

Linear Constructions

Name	Usage Count
Total:	58,236
OA with Only One Run / Dual of Code Without Redundancy	16
OOA with Only One Run / Dual of NRT-Code Without Redundancy	40
Arbitrary OA / Dual of Arbitrary Code	2,187
Arbitrary Net	2,203
Arbitrary OOA / Dual of Arbitrary NRT-Code	6,078
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 Code	464
Reed–Solomon Codes for OOAs	1,564
Extended Reed–Solomon Code	448
Extended Reed–Solomon Codes for OOAs	1,581
Complete OA / Dual of Code with Only One Code Word	2,187
Complete OOA / Dual of NRT-Code with Only One Code Word	2,124
Repetition Code	2,203
Repetition NRT-Code	2,092
Repetition Code with Arbitrary Length	16
Repetition NRT-Code with Arbitrary Length	40
OOA with Large Number of Runs	16
Large High-Dimensional Nets with Fixed Strength	2,187
Explicit Generator Matrices for OOAs by Edel	5
OOAs Constructed from BCH Codes	45
Parity-Check Code	16
Hamming Code	2,171
Simplex Code	28
Reed–Muller Code	53
Extended Quadratic Residue Code	42
Codes of Belov Type	457
Caps in Base b = 2	258
Oval	8
Hyperoval	7
Ovoid	15
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 Hyperplane	32
Caps Completed Using a Computer Search	19
Codes from Dual Arcs	29
Union of Pairwise Disjoint Baer Subplanes	13
Hermitian Unital	3
Projective Codes Based on the Barlotti Arcs	6
Denniston Codes	8
Projective Code from Ovoid	6
Codes by Danev and Olsson	14
Codes by De Boer and Brouwer	56
Augmented Codes by De Boer and Brouwer	12
BCH-Codes	4,545
Primitive BCH-Codes	7,390
Extended Narrow-Sense BCH-Codes	1,617
Primitive Extended Narrow-Sense BCH-Codes	2,501
Primitive Narrow-Sense BCH-Codes (Sporadic Result)	3
Primitive Expurgated Narrow-Sense BCH-Codes (Sporadic Result)	3
Narrow-Sense BCH-Codes	1,337
Primitive Narrow-Sense BCH-Codes	638
Expurgated Narrow-Sense BCH-Codes	496
Primitive Expurgated Narrow-Sense BCH-Codes	980
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 Homepage	29
Quaternary Constacyclic Codes with Minimum Distance 5	21
Ternary Negacyclic Codes with Minimum Distance 5	8
Extended or Lengthened BCH Codes	42
Extended or Lengthened Subfield Codes of Reed–Solomon Codes	9
Codes Obtained from BCH Codes, Concatenation and Construction X	35
Linear Codes Obtained from Additive Codes and Concatenation	25
BCH Codes Lengthened with UEP Codes	33
Codes Constructed by Inverting Construction Y1	43
Extended or Lengthened BCH Codes (Additions)	26
BCH Codes Lengthened with UEP Codes (Additions)	2
Code Obtained from BCH Codes with Construction X4	2
Quasi-Cyclic and Quasi-Twisted Codes by Boukliev	3
Quasi-Cyclic Codes by Boukliev, Daskalov, and Kapralov	2
Quasi-Cyclic Codes by Gulliver and Bhargava	66
Quasi-Cyclic Codes by Bierbrauer and Gulliver	23
Quaternary Quasi-Twisted Codes of Dimension 6 by Gulliver and Östergård	18
Quasi-Cyclic and Quasi-Twisted Codes by Daskalov and Gulliver	135
Cyclic, Quasi-Cyclic, and Quasi-Twisted Codes by Daskalov, Hristov, and Metodieva	165
Codes by Kohnert and Zwanzger	35
Golay Code	2
Extended Golay Code	2
Tetracode	1
Hexacode	1
Hill Cap	1
Glynn Cap	1
A Cap Found by Tallini	1
A Cyclic Code with Given Generator	5
Linear Codes with Explicit Generator Matrix	62
A Code by Marcugini, Milani, and Pambianco	2
A Code by Bierbrauer, Marcugini, and Pambianco	3
k = 2 Construction	2,171
Nets Constructed from Net-Embeddable BCH Codes	123
Sobol Sequence (Bratley–Fox Implementation) with Equidistant Coordinate	4
Niederreiter Sequence (Bratley–Fox–Niederreiter Implementation)	10
Niederreiter Sequence (Bratley–Fox–Niederreiter Implementation) with Equidistant Coordinate	211
Niederreiter–Xing Sequence (Piršić Implementation)	3
Niederreiter–Xing Sequence (Piršić Implementation) with Equidistant Coordinate	245
Shift-Nets	135
Hand-Crafted Matrix by Schmid and Wolf	2
Linear Code Embeddings Found in a Computer Search	129
Salzburg Tables	30
Van der Corput Sequence	35
Faure Sequence	4
Generalized Faure Sequence	31
Sobol Sequence	2
Niederreiter Sequence	222
Arbitrary (t, s)-Sequence for Large s	32
Niederreiter–Xing Sequence Construction III with García–Stichtenoth Tower As Constant Field Extension	106
Construction for Strength k = 1	2,187

Non-Linear Constructions

Name	Usage Count
Total:	24
Nordstrom–Robinson OA	1
Kerdock Codes	10
Delsarte–Goethals Codes	10
Preparata Codes	3

Linear Existence Results

Name	Usage Count
Total:	249,255
Gilbert–Varšamov Bound for OAs	49,924
Gilbert–Varšamov Bound for Nets	42,225
Gilbert–Varšamov Bound for OOAs	152,111
Code from Brouwer’s Database	1,070
Code from Brouwer’s Database	2
Code from Grassl’s Database	2,808
Code from Grassl’s Database	1,109
Plane (s,3)-Arcs in PG(2,b)	2
Existence of Nets from Net-Embeddable BCH Codes	4

Bounds (for the Linear and Non-Linear Case)

Name	Usage Count
Total:	402,499
Trivial Upper Bound on the Number of Factors	8,812
Trivial Upper Bound on s	6,593
Trivial Upper Bound on the Number of Factors for OOAs	32,883
Singleton Bound	2,251
Trivial Lower Bound on t	2,219
Singleton Bound for OOAs	21,846
Trivial Lower Bound on t for Sequences	32
Lower Bound on the Number of Runs for OAs with Strength k = 2 and Arbitrarily Large Number of Factors s	32
Lower Bound on t for Nets with Strength k = 2 and Arbitrarily Large Dimension s	16
Lower Bound on the Number of Runs for OOAs with Strength k = 2 and Arbitrarily Large Number of Factors s	80
Rao or (Dual) Hamming Bound	52,374
Rao or (Dual) Hamming Bound (Large M)	2,062
Generalized Rao Bound for Nets	53,905
Generalized Rao Bound for OOAs	196,401
(Dual) Plotkin Bound	1,061
Generalized (Dual) Plotkin Bound for OOAs	4,338
(Dual) Plotkin Bound for n = 2	4,438
Generalized (Dual) Plotkin Bound for n = 2	130
LP Bound with Quadratic Polynomials for OOAs	103
Bound Derived from the LP Bound by Trinker	24
Linear Programming Bound	5,291
Linear Programming Bound for OOAs	782
Bound for OAs with Strength k = 2	2,171
Bose–Bush Bound for OAs with Strength 3	69
Bound for OAs with Index Unity	14
Bound by Seiden and Zemach	1
Bound by Seiden	1
Trivial Upper Bound on m	2,399
Mutually Orthogonal Hypercube Bound	2,171

Bounds Applicable Only to the Linear Case

Name	Usage Count
Total:	21,971
Trivial Bound on the Size of Affine Caps	2,095
Affine Subcap of Ovoid Is Optimal	15
Sporadic Bounds on the Size of Affine Caps	3
Griesmer Bound	3,023
Johnson Bound	14,244
Improved Johnson Bound	2,170
Linear Programming Bound with Additional Constraints on the Weights	19
Bound for MDS Codes	80
Bound for Almost-MDS-Codes	5
Bound for Almost-MDS-Codes with Dimension n = 3	12
Sharpened Johnson Bound	16
Improvement by Maruta on the Griesmer Bound	10
Ovoid Cap Is Optimal	15
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 Sziklai	2
A Result by Gronchi and Heim	1
A Result by Barát, Edel, Hill, and Storme	1
A Result by Landjev, Maruta, and Hill	1
A Result by Dodunekov and Landjev	2
A Result by Hill and Landjev	2
A Result by Hill	1
A Result by Boukliev, Kapralov, Maruta, and Fukui	4
Bound by Simonis	1
Bound by Fontaine and Peterson	1
Bound on Codes from Brouwer’s Database	236
Bound on Codes from Brouwer’s Database	0
A Bound by Schmid and Wolf	1

Algebraic Function Fields

Name	Usage Count
Total:	987
Rational Function Field	35
Hermitian Function Field	7
Suzuki Function Field	2
Klein Quartic	1
An Explicitly Constructive Algebraic Function Field	88
Function Fields by García and Quoos	26
Function Fields by Sémirat	48
Sharp Bound on the Number of Rational Points on Curves with Genus 1	6
Sharp Bound on the Number of Rational Points on Curves with Genus 2	9
Fields by Serre with Genus 3 or 4	2
Drinfeld Modules of Rank 1	245
Algebraic Function Fields over ℤ3 by Niederreiter/Xing	24
Algebraic Function Fields over ℤ5 by Niederreiter/Xing	51
Algebraic Function Fields over F9 and F27 by Niederreiter/Xing	1
Algebraic Function Fields by Auer	11
Algebraic Function Fields by Keller	3
Algebraic Function Fields by Beelen and Pellikaan	5
Algebraic Function Fields by Teo	19
Algebraic Function Fields over ℤ7 by Yeo	5
ManYPoints Database	319
Table by Niederreiter and Xing	7
Tower of Function Fields by García and Stichtenoth	9
Second Tower of Function Fields by García and Stichtenoth	16
Tower of Function Fields by Niederreiter and Xing Based on the Tower by García and Stichtenoth	8
Tower of Function Fields by García, Stichtenoth, and Rück	13
Tower of Function Fields by Bezerra and García	17
Tower of Function Fields over F8 by van der Geer and van der Vlugt	5
Tower of Function Fields by Bezerra, García, and Stichtenoth	5

Propagation Rules

Name	Usage Count
Total:	2,600,827
Overall Total:	3,333,799
Strength Reduction	54,057
t-Expansion	116,363
Strength Reduction for OOAs	241,698
t-Expansion for Sequences	3,452
Code Embedding in Larger Space	45,475
NRT-Code Embedding in Larger Space	55,790
Duplication / Taking a Subcode of the Dual Code	2,171
Net Duplication	26,506
OOA Duplication	142,243
Truncation	187,666
Truncation for OOAs	290,449
m-Reduction	177,475
m-Reduction from Arbitrarily Large Net	2,406
m-Reduction for OOAs	81,267
Repeating Each Code Word	5,188
Repeating Each Code Word for NRT-Codes	12,194
Trace Code	5,643
Trace Code for Nets	20,114
Trace Code for OOAs	4,947
Base Expansion for Nets	0
Base Expansion for OOAs	45
Base Expansion for Sequences	9
Base Reduction	0
Base Reduction for Sequences	306
Base Reduction for Projective Spaces	28
Base Reduction for Projective Spaces (for Nets)	326
Discarding Parts of the Base for OAs	8,378
Discarding Parts of the Base for OOAs	2,162
Direct Product of Two Codes	0
Direct Product of Two Nets	0
Direct Product of Two OOAs	0
(u, u+v)-Construction	11,815
(u, u+v)-Construction for Nets	26,197
(u, u+v)-Construction for OOAs	44,493
Generalized (u, u+v)-Construction	1,005
Generalized (u, u+v)-Construction for Nets	5,177
Generalized (u, u+v)-Construction for OOAs	4,492
Juxtaposition	3,003
Juxtaposition for OOAs	9,064
Concatenation of Two Codes	11,886
Concatenation of Two NRT-Codes	11,500
Concatenation of Two Orthogonal Arrays	0
Concatenation of Two OOAs	0
Construction X with Reed–Solomon Codes	1,337
Construction X with Algebraic-Geometric Codes	17,787
Construction X with Algebraic-Geometric NRT-Codes	67,492
Construction XX with a Chain of Algebraic-Geometric Codes	312
Construction X with OOAs from (t, s)-Sequence	279
Depth Reduction	170,049
Extracting Embedded Orthogonal Array	5,262
Extracting Embedded OOA	46,782
Appending Arbitrary Columns	7,004
Net Defined by OOA	35,287
Embedding of OOA with Gilbert–Varšamov Bound	82,123
Niederreiter–Xing Sequence Construction II/III	515
Niederreiter–Xing Sequence Construction III Using Non-Rational Places	26
Extended Algebraic-Geometric Codes	8,777
Extended Algebraic-Geometric NRT-Codes	64,023
Algebraic-Geometric Codes	4,583
Algebraic-Geometric NRT-Codes	13,769
Algebraic-Geometric Codes with Known Gap Numbers	19
Algebraic-Geometric NRT-Codes with Known Gap Numbers	32
Algebraic-Geometric Codes Defined Using a Non-Rational Place	399
Algebraic-Geometric NRT-Codes Defined Using a Non-Rational Place	615
A Recursive Bound on the Size of Affine Caps by Bierbrauer and Edel	53
Base Reduction for Affine Caps	3
Every Affine Cap Is Also a Projective Cap	18
Doubling a Cap (and Obtaining an Affine Cap)	11
Adding a Parity Check Bit	1,331
Residual Code	1,828
Dual of MDS Code Is Again an MDS Code	14
Dual of a Near-MDS Code Is Again a Near-MDS Code	13
Contraction	23
Contraction (with Narrow-Sense BCH-Code)	57
Contraction (with Expurgated Narrow-Sense BCH-Code)	63
(u, u−v, u+v+w)-Construction	2,251
Construction Y1 (Bound)	536
Construction Y1	69
Dual Code (with Bound on d by Construction Y1)	6
Varšamov–Edel Lengthening	12,752
Removing Affine Subspaces	38
Hill Recurrence	5
Doubling a Cap	10
Product of Projective Cap with (Hyper)oval	9
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)oval	2
Product of Two Projective Caps and an Affine Cap Avoiding Two Hyperplanes	2
Construction for [s, s−6, 6]-Code from [s, 3, s−3]-Code	4
Construction X with Cyclic Codes	9,011
Construction XX with Cyclic Codes	16,740
Construction XX with a Chain of Cyclic Codes	788
Construction X4 with Cyclic Codes	155
Construction X with Extended Narrow-Sense BCH Codes	17,255
Construction XX with a Chain of Extended Narrow-Sense BCH Codes	3,895
Construction X4 with Extended Narrow-Sense BCH Codes	424
Construction X with De Boer–Brouwer Codes	201
Construction XX with De Boer–Brouwer Codes	9
Construction XX with a Chain of De Boer–Brouwer Codes	181
Construction X with Varšamov Bound	4,988
Construction X with Varšamov–Edel Bound	19
Construction X with Ovoid	11
Construction X with Kerdock and Delsarte–Goethals OAs	19
Construction XX with a Chain of Delsarte–Goethals OAs	0
Construction X4 with Kerdock OAs	17
Standard Lengthening for NRT-Codes	32
Net from Sequence	81,788
OOA from Sequence	174
Logical Equivalence (for Sequences)	9,548
Logical Equivalence (for Nets with Unbounded m)	9,548
Base Change	2,825
Appending kth Column	1,251
OAs with Strength 3, b ≠ 2, and m > 3 Are Always Embeddable	123
OOA Folding	239,077
OA Folding and Stacking	4,852
OOA Folding and Stacking with Additional Row	26,774
OOA Stacking with Additional Row	10,530

Other Descriptions

• Discarding Factors / Shortening the Dual Code
• s-Reduction
• Discarding Factors for OOAs / Shortening the Dual NRT-Code
• s-Reduction for Sequences
• Construction X
• Construction X4
• Product of Cap in PG with Cap in AG
• Restricted Johnson Bound