Best Known (180−36, 180, s)-Nets in Base 2
(180−36, 180, 260)-Net over F2 — Constructive and digital
Digital (144, 180, 260)-net over F2, using
- 4 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
(180−36, 180, 507)-Net over F2 — Digital
Digital (144, 180, 507)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2180, 507, F2, 2, 36) (dual of [(507, 2), 834, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2180, 521, F2, 2, 36) (dual of [(521, 2), 862, 37]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2179, 521, F2, 2, 36) (dual of [(521, 2), 863, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2179, 1042, F2, 36) (dual of [1042, 863, 37]-code), using
- strength reduction [i] based on linear OA(2179, 1042, F2, 37) (dual of [1042, 863, 38]-code), using
- adding a parity check bit [i] based on linear OA(2178, 1041, F2, 36) (dual of [1041, 863, 37]-code), using
- construction XX applied to C1 = C([1021,32]), C2 = C([1,34]), C3 = C1 + C2 = C([1,32]), and C∩ = C1 ∩ C2 = C([1021,34]) [i] based on
- linear OA(2171, 1023, F2, 35) (dual of [1023, 852, 36]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,32}, and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(2165, 1023, F2, 34) (dual of [1023, 858, 35]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [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 = {−2,−1,…,34}, and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(2160, 1023, F2, 32) (dual of [1023, 863, 33]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- 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
- linear OA(21, 6, F2, 1) (dual of [6, 5, 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 (see above)
- construction XX applied to C1 = C([1021,32]), C2 = C([1,34]), C3 = C1 + C2 = C([1,32]), and C∩ = C1 ∩ C2 = C([1021,34]) [i] based on
- adding a parity check bit [i] based on linear OA(2178, 1041, F2, 36) (dual of [1041, 863, 37]-code), using
- strength reduction [i] based on linear OA(2179, 1042, F2, 37) (dual of [1042, 863, 38]-code), using
- OOA 2-folding [i] based on linear OA(2179, 1042, F2, 36) (dual of [1042, 863, 37]-code), using
- 21 times duplication [i] based on linear OOA(2179, 521, F2, 2, 36) (dual of [(521, 2), 863, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2180, 521, F2, 2, 36) (dual of [(521, 2), 862, 37]-NRT-code), using
(180−36, 180, 7708)-Net in Base 2 — Upper bound on s
There is no (144, 180, 7709)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1 536060 621593 447026 727309 179041 391617 668996 254066 253570 > 2180 [i]