Best Known (188−33, 188, s)-Nets in Base 2
(188−33, 188, 320)-Net over F2 — Constructive and digital
Digital (155, 188, 320)-net over F2, using
- 2 times m-reduction [i] based on digital (155, 190, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 38, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- trace code for nets [i] based on digital (3, 38, 64)-net over F32, using
(188−33, 188, 843)-Net over F2 — Digital
Digital (155, 188, 843)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2188, 843, F2, 2, 33) (dual of [(843, 2), 1498, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2188, 1042, F2, 2, 33) (dual of [(1042, 2), 1896, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2188, 2084, F2, 33) (dual of [2084, 1896, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(2188, 2085, F2, 33) (dual of [2085, 1897, 34]-code), using
- construction XX applied to Ce(32) ⊂ Ce(28) ⊂ Ce(26) [i] based on
- linear OA(2177, 2048, F2, 33) (dual of [2048, 1871, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2155, 2048, F2, 29) (dual of [2048, 1893, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2144, 2048, F2, 27) (dual of [2048, 1904, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- linear OA(21, 5, F2, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction XX applied to Ce(32) ⊂ Ce(28) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(2188, 2085, F2, 33) (dual of [2085, 1897, 34]-code), using
- OOA 2-folding [i] based on linear OA(2188, 2084, F2, 33) (dual of [2084, 1896, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(2188, 1042, F2, 2, 33) (dual of [(1042, 2), 1896, 34]-NRT-code), using
(188−33, 188, 22406)-Net in Base 2 — Upper bound on s
There is no (155, 188, 22407)-net in base 2, because
- 1 times m-reduction [i] would yield (155, 187, 22407)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 196 245731 925918 458229 033139 464708 998720 122024 209792 751400 > 2187 [i]