Best Known (167−34, 167, s)-Nets in Base 2
(167−34, 167, 260)-Net over F2 — Constructive and digital
Digital (133, 167, 260)-net over F2, using
- t-expansion [i] based on digital (132, 167, 260)-net over F2, using
- 1 times m-reduction [i] based on digital (132, 168, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 42, 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, 42, 65)-net over F16, using
- 1 times m-reduction [i] based on digital (132, 168, 260)-net over F2, using
(167−34, 167, 454)-Net over F2 — Digital
Digital (133, 167, 454)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2167, 454, F2, 2, 34) (dual of [(454, 2), 741, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2167, 515, F2, 2, 34) (dual of [(515, 2), 863, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2167, 1030, F2, 34) (dual of [1030, 863, 35]-code), using
- strength reduction [i] based on linear OA(2167, 1030, F2, 35) (dual of [1030, 863, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(32) [i] based on
- linear OA(2166, 1024, F2, 35) (dual of [1024, 858, 36]-code), using an extension Ce(34) of 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(2161, 1024, F2, 33) (dual of [1024, 863, 34]-code), using an extension Ce(32) of 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, 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, using
- construction X applied to Ce(34) ⊂ Ce(32) [i] based on
- strength reduction [i] based on linear OA(2167, 1030, F2, 35) (dual of [1030, 863, 36]-code), using
- OOA 2-folding [i] based on linear OA(2167, 1030, F2, 34) (dual of [1030, 863, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(2167, 515, F2, 2, 34) (dual of [(515, 2), 863, 35]-NRT-code), using
(167−34, 167, 6478)-Net in Base 2 — Upper bound on s
There is no (133, 167, 6479)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 187 478331 871025 296939 890074 741976 327052 213649 778752 > 2167 [i]