Best Known (216−37, 216, s)-Nets in Base 2
(216−37, 216, 380)-Net over F2 — Constructive and digital
Digital (179, 216, 380)-net over F2, using
- 21 times duplication [i] based on digital (178, 215, 380)-net over F2, using
- t-expansion [i] based on digital (177, 215, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 43, 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, 43, 76)-net over F32, using
- t-expansion [i] based on digital (177, 215, 380)-net over F2, using
(216−37, 216, 1013)-Net over F2 — Digital
Digital (179, 216, 1013)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2216, 1013, F2, 2, 37) (dual of [(1013, 2), 1810, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2216, 1049, F2, 2, 37) (dual of [(1049, 2), 1882, 38]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2212, 1047, F2, 2, 37) (dual of [(1047, 2), 1882, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2212, 2094, F2, 37) (dual of [2094, 1882, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(30) [i] based on
- linear OA(2199, 2048, F2, 37) (dual of [2048, 1849, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(2166, 2048, F2, 31) (dual of [2048, 1882, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(213, 46, F2, 5) (dual of [46, 33, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- construction X applied to Ce(36) ⊂ Ce(30) [i] based on
- OOA 2-folding [i] based on linear OA(2212, 2094, F2, 37) (dual of [2094, 1882, 38]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2212, 1047, F2, 2, 37) (dual of [(1047, 2), 1882, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2216, 1049, F2, 2, 37) (dual of [(1049, 2), 1882, 38]-NRT-code), using
(216−37, 216, 29742)-Net in Base 2 — Upper bound on s
There is no (179, 216, 29743)-net in base 2, because
- 1 times m-reduction [i] would yield (179, 215, 29743)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 52667 643942 368249 428829 867876 240094 167070 629259 946046 565604 902374 > 2215 [i]