Best Known (215, 215+45, s)-Nets in Base 2
(215, 215+45, 490)-Net over F2 — Constructive and digital
Digital (215, 260, 490)-net over F2, using
- trace code for nets [i] based on digital (7, 52, 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
(215, 215+45, 1049)-Net over F2 — Digital
Digital (215, 260, 1049)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2260, 1049, F2, 2, 45) (dual of [(1049, 2), 1838, 46]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2256, 1047, F2, 2, 45) (dual of [(1047, 2), 1838, 46]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2256, 2094, F2, 45) (dual of [2094, 1838, 46]-code), using
- construction X applied to Ce(44) ⊂ Ce(38) [i] based on
- linear OA(2243, 2048, F2, 45) (dual of [2048, 1805, 46]-code), using an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(2210, 2048, F2, 39) (dual of [2048, 1838, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [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(44) ⊂ Ce(38) [i] based on
- OOA 2-folding [i] based on linear OA(2256, 2094, F2, 45) (dual of [2094, 1838, 46]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2256, 1047, F2, 2, 45) (dual of [(1047, 2), 1838, 46]-NRT-code), using
(215, 215+45, 31648)-Net in Base 2 — Upper bound on s
There is no (215, 260, 31649)-net in base 2, because
- 1 times m-reduction [i] would yield (215, 259, 31649)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 926733 803261 124516 458337 378388 845485 948530 849064 425652 369719 718181 441955 559184 > 2259 [i]