Best Known (132, 132+34, s)-Nets in Base 2
(132, 132+34, 260)-Net over F2 — Constructive and digital
Digital (132, 166, 260)-net over F2, using
- 2 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
(132, 132+34, 443)-Net over F2 — Digital
Digital (132, 166, 443)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2166, 443, F2, 2, 34) (dual of [(443, 2), 720, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2166, 514, F2, 2, 34) (dual of [(514, 2), 862, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2166, 1028, F2, 34) (dual of [1028, 862, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(2166, 1029, F2, 34) (dual of [1029, 863, 35]-code), using
- 1 times truncation [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
- 1 times truncation [i] based on linear OA(2167, 1030, F2, 35) (dual of [1030, 863, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2166, 1029, F2, 34) (dual of [1029, 863, 35]-code), using
- OOA 2-folding [i] based on linear OA(2166, 1028, F2, 34) (dual of [1028, 862, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(2166, 514, F2, 2, 34) (dual of [(514, 2), 862, 35]-NRT-code), using
(132, 132+34, 6218)-Net in Base 2 — Upper bound on s
There is no (132, 166, 6219)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 93 701605 352335 346890 927368 499648 604074 560980 357552 > 2166 [i]