Best Known (237−37, 237, s)-Nets in Base 2
(237−37, 237, 520)-Net over F2 — Constructive and digital
Digital (200, 237, 520)-net over F2, using
- 22 times duplication [i] based on digital (198, 235, 520)-net over F2, using
- t-expansion [i] based on digital (197, 235, 520)-net over F2, using
- trace code for nets [i] based on digital (9, 47, 104)-net over F32, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- trace code for nets [i] based on digital (9, 47, 104)-net over F32, using
- t-expansion [i] based on digital (197, 235, 520)-net over F2, using
(237−37, 237, 1943)-Net over F2 — Digital
Digital (200, 237, 1943)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2237, 1943, F2, 4, 37) (dual of [(1943, 4), 7535, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2237, 2051, F2, 4, 37) (dual of [(2051, 4), 7967, 38]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2236, 2051, F2, 4, 37) (dual of [(2051, 4), 7968, 38]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2236, 8204, F2, 37) (dual of [8204, 7968, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(2236, 8206, F2, 37) (dual of [8206, 7970, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- linear OA(2235, 8192, F2, 37) (dual of [8192, 7957, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(2222, 8192, F2, 35) (dual of [8192, 7970, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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(36) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(2236, 8206, F2, 37) (dual of [8206, 7970, 38]-code), using
- OOA 4-folding [i] based on linear OA(2236, 8204, F2, 37) (dual of [8204, 7968, 38]-code), using
- 21 times duplication [i] based on linear OOA(2236, 2051, F2, 4, 37) (dual of [(2051, 4), 7968, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2237, 2051, F2, 4, 37) (dual of [(2051, 4), 7967, 38]-NRT-code), using
(237−37, 237, 66802)-Net in Base 2 — Upper bound on s
There is no (200, 237, 66803)-net in base 2, because
- 1 times m-reduction [i] would yield (200, 236, 66803)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 110429 786940 387422 979778 063360 681271 650798 279769 641010 854510 281907 548199 > 2236 [i]