Best Known (56−18, 56, s)-Nets in Base 2
(56−18, 56, 54)-Net over F2 — Constructive and digital
Digital (38, 56, 54)-net over F2, using
- trace code for nets [i] based on digital (10, 28, 27)-net over F4, using
- net from sequence [i] based on digital (10, 26)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 10 and N(F) ≥ 27, using
- net from sequence [i] based on digital (10, 26)-sequence over F4, using
(56−18, 56, 60)-Net over F2 — Digital
Digital (38, 56, 60)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(256, 60, F2, 2, 18) (dual of [(60, 2), 64, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(256, 63, F2, 2, 18) (dual of [(63, 2), 70, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(256, 126, F2, 18) (dual of [126, 70, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(256, 127, F2, 18) (dual of [127, 71, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 127 = 27−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(256, 127, F2, 18) (dual of [127, 71, 19]-code), using
- OOA 2-folding [i] based on linear OA(256, 126, F2, 18) (dual of [126, 70, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(256, 63, F2, 2, 18) (dual of [(63, 2), 70, 19]-NRT-code), using
(56−18, 56, 296)-Net in Base 2 — Upper bound on s
There is no (38, 56, 297)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 72358 359571 402378 > 256 [i]