Best Known (222−33, 222, s)-Nets in Base 2
(222−33, 222, 624)-Net over F2 — Constructive and digital
Digital (189, 222, 624)-net over F2, using
- t-expansion [i] based on digital (188, 222, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 37, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 37, 104)-net over F64, using
(222−33, 222, 2148)-Net over F2 — Digital
Digital (189, 222, 2148)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2222, 2148, F2, 3, 33) (dual of [(2148, 3), 6222, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2222, 2748, F2, 3, 33) (dual of [(2748, 3), 8022, 34]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2222, 8244, F2, 33) (dual of [8244, 8022, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(26) [i] based on
- linear OA(2209, 8192, F2, 33) (dual of [8192, 7983, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2170, 8192, F2, 27) (dual of [8192, 8022, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(213, 52, F2, 5) (dual of [52, 39, 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(32) ⊂ Ce(26) [i] based on
- OOA 3-folding [i] based on linear OA(2222, 8244, F2, 33) (dual of [8244, 8022, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(2222, 2748, F2, 3, 33) (dual of [(2748, 3), 8022, 34]-NRT-code), using
(222−33, 222, 97816)-Net in Base 2 — Upper bound on s
There is no (189, 222, 97817)-net in base 2, because
- 1 times m-reduction [i] would yield (189, 221, 97817)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3 370379 769258 172784 796026 686197 220202 404047 489538 635975 205000 411599 > 2221 [i]