Best Known (147, 184, s)-Nets in Base 2
(147, 184, 260)-Net over F2 — Constructive and digital
Digital (147, 184, 260)-net over F2, using
- 4 times m-reduction [i] based on digital (147, 188, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 47, 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, 47, 65)-net over F16, using
(147, 184, 506)-Net over F2 — Digital
Digital (147, 184, 506)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2184, 506, F2, 2, 37) (dual of [(506, 2), 828, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2184, 528, F2, 2, 37) (dual of [(528, 2), 872, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2184, 1056, F2, 37) (dual of [1056, 872, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(2184, 1057, F2, 37) (dual of [1057, 873, 38]-code), using
- adding a parity check bit [i] based on linear OA(2183, 1056, F2, 36) (dual of [1056, 873, 37]-code), using
- construction XX applied to C1 = C([989,1022]), C2 = C([993,2]), C3 = C1 + C2 = C([993,1022]), and C∩ = C1 ∩ C2 = C([989,2]) [i] based on
- linear OA(2165, 1023, F2, 34) (dual of [1023, 858, 35]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−34,−33,…,−1}, and designed minimum distance d ≥ |I|+1 = 35 [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(2150, 1023, F2, 30) (dual of [1023, 873, 31]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−30,−29,…,−1}, and designed minimum distance d ≥ |I|+1 = 31 [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, 12, F2, 1) (dual of [12, 11, 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([989,1022]), C2 = C([993,2]), C3 = C1 + C2 = C([993,1022]), and C∩ = C1 ∩ C2 = C([989,2]) [i] based on
- adding a parity check bit [i] based on linear OA(2183, 1056, F2, 36) (dual of [1056, 873, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(2184, 1057, F2, 37) (dual of [1057, 873, 38]-code), using
- OOA 2-folding [i] based on linear OA(2184, 1056, F2, 37) (dual of [1056, 872, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(2184, 528, F2, 2, 37) (dual of [(528, 2), 872, 38]-NRT-code), using
(147, 184, 8655)-Net in Base 2 — Upper bound on s
There is no (147, 184, 8656)-net in base 2, because
- 1 times m-reduction [i] would yield (147, 183, 8656)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 12 280955 750811 237831 784256 402703 188897 506817 555436 580700 > 2183 [i]