Best Known (219−37, 219, s)-Nets in Base 2
(219−37, 219, 380)-Net over F2 — Constructive and digital
Digital (182, 219, 380)-net over F2, using
- t-expansion [i] based on digital (181, 219, 380)-net over F2, using
- 1 times m-reduction [i] based on digital (181, 220, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 44, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- trace code for nets [i] based on digital (5, 44, 76)-net over F32, using
- 1 times m-reduction [i] based on digital (181, 220, 380)-net over F2, using
(219−37, 219, 1182)-Net over F2 — Digital
Digital (182, 219, 1182)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2219, 1182, F2, 3, 37) (dual of [(1182, 3), 3327, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2219, 1370, F2, 3, 37) (dual of [(1370, 3), 3891, 38]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2219, 4110, F2, 37) (dual of [4110, 3891, 38]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2218, 4109, F2, 37) (dual of [4109, 3891, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- linear OA(2217, 4096, F2, 37) (dual of [4096, 3879, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(21, 13, F2, 1) (dual of [13, 12, 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
- 1 times code embedding in larger space [i] based on linear OA(2218, 4109, F2, 37) (dual of [4109, 3891, 38]-code), using
- OOA 3-folding [i] based on linear OA(2219, 4110, F2, 37) (dual of [4110, 3891, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(2219, 1370, F2, 3, 37) (dual of [(1370, 3), 3891, 38]-NRT-code), using
(219−37, 219, 33388)-Net in Base 2 — Upper bound on s
There is no (182, 219, 33389)-net in base 2, because
- 1 times m-reduction [i] would yield (182, 218, 33389)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 421424 805024 886106 828114 653191 018146 439233 835344 468884 351151 229764 > 2218 [i]