Best Known (223−33, 223, s)-Nets in Base 2
(223−33, 223, 624)-Net over F2 — Constructive and digital
Digital (190, 223, 624)-net over F2, using
- 21 times duplication [i] based on 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
- t-expansion [i] based on digital (188, 222, 624)-net over F2, using
(223−33, 223, 2201)-Net over F2 — Digital
Digital (190, 223, 2201)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2223, 2201, F2, 3, 33) (dual of [(2201, 3), 6380, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2223, 2748, F2, 3, 33) (dual of [(2748, 3), 8021, 34]-NRT-code), using
- 21 times duplication [i] 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
- 21 times duplication [i] based on linear OOA(2222, 2748, F2, 3, 33) (dual of [(2748, 3), 8022, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2223, 2748, F2, 3, 33) (dual of [(2748, 3), 8021, 34]-NRT-code), using
(223−33, 223, 102148)-Net in Base 2 — Upper bound on s
There is no (190, 223, 102149)-net in base 2, because
- 1 times m-reduction [i] would yield (190, 222, 102149)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 6 741003 608433 140054 929754 686724 307942 302783 804476 934181 266581 118160 > 2222 [i]