Best Known (213−35, 213, s)-Nets in Base 2
(213−35, 213, 490)-Net over F2 — Constructive and digital
Digital (178, 213, 490)-net over F2, using
- 23 times duplication [i] based on digital (175, 210, 490)-net over F2, using
- trace code for nets [i] based on digital (7, 42, 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, 42, 98)-net over F32, using
(213−35, 213, 1314)-Net over F2 — Digital
Digital (178, 213, 1314)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2213, 1314, F2, 3, 35) (dual of [(1314, 3), 3729, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2213, 1376, F2, 3, 35) (dual of [(1376, 3), 3915, 36]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2213, 4128, F2, 35) (dual of [4128, 3915, 36]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2211, 4126, F2, 35) (dual of [4126, 3915, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- 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(2181, 4096, F2, 31) (dual of [4096, 3915, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(34) ⊂ Ce(30) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2211, 4126, F2, 35) (dual of [4126, 3915, 36]-code), using
- OOA 3-folding [i] based on linear OA(2213, 4128, F2, 35) (dual of [4128, 3915, 36]-code), using
- discarding factors / shortening the dual code based on linear OOA(2213, 1376, F2, 3, 35) (dual of [(1376, 3), 3915, 36]-NRT-code), using
(213−35, 213, 40709)-Net in Base 2 — Upper bound on s
There is no (178, 213, 40710)-net in base 2, because
- 1 times m-reduction [i] would yield (178, 212, 40710)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 6582 357601 956703 081807 743015 760974 144037 001682 010938 859873 774000 > 2212 [i]