Best Known (116, 116+27, s)-Nets in Base 2
(116, 116+27, 260)-Net over F2 — Constructive and digital
Digital (116, 143, 260)-net over F2, using
- t-expansion [i] based on digital (114, 143, 260)-net over F2, using
- 1 times m-reduction [i] based on digital (114, 144, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 36, 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, 36, 65)-net over F16, using
- 1 times m-reduction [i] based on digital (114, 144, 260)-net over F2, using
(116, 116+27, 537)-Net over F2 — Digital
Digital (116, 143, 537)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2143, 537, F2, 2, 27) (dual of [(537, 2), 931, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2143, 1074, F2, 27) (dual of [1074, 931, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2143, 1075, F2, 27) (dual of [1075, 932, 28]-code), using
- construction XX applied to C1 = C([1019,18]), C2 = C([0,22]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([1019,22]) [i] based on
- linear OA(2111, 1023, F2, 23) (dual of [1023, 912, 24]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−4,−3,…,18}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(2111, 1023, F2, 23) (dual of [1023, 912, 24]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(2131, 1023, F2, 27) (dual of [1023, 892, 28]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−4,−3,…,22}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(291, 1023, F2, 19) (dual of [1023, 932, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(26, 26, F2, 3) (dual of [26, 20, 4]-code or 26-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(26, 26, F2, 3) (dual of [26, 20, 4]-code or 26-cap in PG(5,2)) (see above)
- construction XX applied to C1 = C([1019,18]), C2 = C([0,22]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([1019,22]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2143, 1075, F2, 27) (dual of [1075, 932, 28]-code), using
- OOA 2-folding [i] based on linear OA(2143, 1074, F2, 27) (dual of [1074, 931, 28]-code), using
(116, 116+27, 10985)-Net in Base 2 — Upper bound on s
There is no (116, 143, 10986)-net in base 2, because
- 1 times m-reduction [i] would yield (116, 142, 10986)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 5 576769 593209 744484 728076 393511 117410 038584 > 2142 [i]