Best Known (146, 146+37, s)-Nets in Base 2
(146, 146+37, 260)-Net over F2 — Constructive and digital
Digital (146, 183, 260)-net over F2, using
- t-expansion [i] based on digital (144, 183, 260)-net over F2, using
- 1 times m-reduction [i] based on digital (144, 184, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 46, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- trace code for nets [i] based on digital (6, 46, 65)-net over F16, using
- 1 times m-reduction [i] based on digital (144, 184, 260)-net over F2, using
(146, 146+37, 495)-Net over F2 — Digital
Digital (146, 183, 495)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2183, 495, F2, 2, 37) (dual of [(495, 2), 807, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2183, 527, F2, 2, 37) (dual of [(527, 2), 871, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2183, 1054, F2, 37) (dual of [1054, 871, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(2183, 1055, F2, 37) (dual of [1055, 872, 38]-code), using
- construction XX applied to C1 = C([989,0]), C2 = C([993,2]), C3 = C1 + C2 = C([993,0]), and C∩ = C1 ∩ C2 = C([989,2]) [i] based on
- linear OA(2166, 1023, F2, 35) (dual of [1023, 857, 36]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−34,−33,…,0}, and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(2161, 1023, F2, 33) (dual of [1023, 862, 34]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−30,−29,…,2}, and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(2176, 1023, F2, 37) (dual of [1023, 847, 38]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−34,−33,…,2}, and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(2151, 1023, F2, 31) (dual of [1023, 872, 32]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−30,−29,…,0}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(26, 21, F2, 3) (dual of [21, 15, 4]-code or 21-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- linear OA(21, 11, F2, 1) (dual of [11, 10, 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 C1 = C([989,0]), C2 = C([993,2]), C3 = C1 + C2 = C([993,0]), and C∩ = C1 ∩ C2 = C([989,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2183, 1055, F2, 37) (dual of [1055, 872, 38]-code), using
- OOA 2-folding [i] based on linear OA(2183, 1054, F2, 37) (dual of [1054, 871, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(2183, 527, F2, 2, 37) (dual of [(527, 2), 871, 38]-NRT-code), using
(146, 146+37, 8327)-Net in Base 2 — Upper bound on s
There is no (146, 183, 8328)-net in base 2, because
- 1 times m-reduction [i] would yield (146, 182, 8328)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 6 140328 503081 326527 962729 117035 998610 296961 498344 359864 > 2182 [i]