Best Known (178−37, 178, s)-Nets in Base 2
(178−37, 178, 260)-Net over F2 — Constructive and digital
Digital (141, 178, 260)-net over F2, using
- 2 times m-reduction [i] based on digital (141, 180, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 45, 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, 45, 65)-net over F16, using
(178−37, 178, 442)-Net over F2 — Digital
Digital (141, 178, 442)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2178, 442, F2, 2, 37) (dual of [(442, 2), 706, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2178, 520, F2, 2, 37) (dual of [(520, 2), 862, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2178, 1040, F2, 37) (dual of [1040, 862, 38]-code), using
- construction XX applied to C1 = C([1021,32]), C2 = C([0,34]), C3 = C1 + C2 = C([0,32]), 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(2166, 1023, F2, 35) (dual of [1023, 857, 36]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,34], and designed minimum distance d ≥ |I|+1 = 36 [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(2161, 1023, F2, 33) (dual of [1023, 862, 34]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,32], and designed minimum distance d ≥ |I|+1 = 34 [i]
- 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
- 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([0,34]), C3 = C1 + C2 = C([0,32]), and C∩ = C1 ∩ C2 = C([1021,34]) [i] based on
- OOA 2-folding [i] based on linear OA(2178, 1040, F2, 37) (dual of [1040, 862, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(2178, 520, F2, 2, 37) (dual of [(520, 2), 862, 38]-NRT-code), using
(178−37, 178, 6864)-Net in Base 2 — Upper bound on s
There is no (141, 178, 6865)-net in base 2, because
- 1 times m-reduction [i] would yield (141, 177, 6865)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 191979 210674 800912 116824 037529 249820 765768 818234 798228 > 2177 [i]