Best Known (236−39, 236, s)-Nets in Base 2
(236−39, 236, 490)-Net over F2 — Constructive and digital
Digital (197, 236, 490)-net over F2, using
- 21 times duplication [i] based on digital (196, 235, 490)-net over F2, using
- t-expansion [i] based on digital (195, 235, 490)-net over F2, using
- trace code for nets [i] based on digital (7, 47, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- trace code for nets [i] based on digital (7, 47, 98)-net over F32, using
- t-expansion [i] based on digital (195, 235, 490)-net over F2, using
(236−39, 236, 1353)-Net over F2 — Digital
Digital (197, 236, 1353)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2236, 1353, F2, 3, 39) (dual of [(1353, 3), 3823, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2236, 1375, F2, 3, 39) (dual of [(1375, 3), 3889, 40]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2235, 1375, F2, 3, 39) (dual of [(1375, 3), 3890, 40]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2235, 4125, F2, 39) (dual of [4125, 3890, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(2235, 4126, F2, 39) (dual of [4126, 3891, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(34) [i] based on
- linear OA(2229, 4096, F2, 39) (dual of [4096, 3867, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(2205, 4096, F2, 35) (dual of [4096, 3891, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(26, 30, F2, 3) (dual of [30, 24, 4]-code or 30-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
- construction X applied to Ce(38) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(2235, 4126, F2, 39) (dual of [4126, 3891, 40]-code), using
- OOA 3-folding [i] based on linear OA(2235, 4125, F2, 39) (dual of [4125, 3890, 40]-code), using
- 21 times duplication [i] based on linear OOA(2235, 1375, F2, 3, 39) (dual of [(1375, 3), 3890, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2236, 1375, F2, 3, 39) (dual of [(1375, 3), 3889, 40]-NRT-code), using
(236−39, 236, 41897)-Net in Base 2 — Upper bound on s
There is no (197, 236, 41898)-net in base 2, because
- 1 times m-reduction [i] would yield (197, 235, 41898)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 55219 042732 874740 877797 550804 090765 211236 901513 929318 953246 510036 518064 > 2235 [i]