Best Known (210−35, 210, s)-Nets in Base 2
(210−35, 210, 490)-Net over F2 — Constructive and digital
Digital (175, 210, 490)-net over F2, using
- trace code for nets [i] based on digital (7, 42, 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
(210−35, 210, 1226)-Net over F2 — Digital
Digital (175, 210, 1226)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2210, 1226, F2, 3, 35) (dual of [(1226, 3), 3468, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2210, 1371, F2, 3, 35) (dual of [(1371, 3), 3903, 36]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2207, 1370, F2, 3, 35) (dual of [(1370, 3), 3903, 36]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2207, 4110, F2, 35) (dual of [4110, 3903, 36]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2206, 4109, F2, 35) (dual of [4109, 3903, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(32) [i] based on
- 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(2193, 4096, F2, 33) (dual of [4096, 3903, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [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(34) ⊂ Ce(32) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2206, 4109, F2, 35) (dual of [4109, 3903, 36]-code), using
- OOA 3-folding [i] based on linear OA(2207, 4110, F2, 35) (dual of [4110, 3903, 36]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2207, 1370, F2, 3, 35) (dual of [(1370, 3), 3903, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2210, 1371, F2, 3, 35) (dual of [(1371, 3), 3903, 36]-NRT-code), using
(210−35, 210, 36019)-Net in Base 2 — Upper bound on s
There is no (175, 210, 36020)-net in base 2, because
- 1 times m-reduction [i] would yield (175, 209, 36020)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 822 802076 885728 668530 249831 581177 529916 623904 045939 732405 149803 > 2209 [i]