Best Known (164, 164+23, s)-Nets in Base 2
(164, 164+23, 5961)-Net over F2 — Constructive and digital
Digital (164, 187, 5961)-net over F2, using
- 23 times duplication [i] based on digital (161, 184, 5961)-net over F2, using
- net defined by OOA [i] based on linear OOA(2184, 5961, F2, 23, 23) (dual of [(5961, 23), 136919, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2184, 65572, F2, 23) (dual of [65572, 65388, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2184, 65575, F2, 23) (dual of [65575, 65391, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(2177, 65536, F2, 23) (dual of [65536, 65359, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2145, 65536, F2, 19) (dual of [65536, 65391, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(27, 39, F2, 3) (dual of [39, 32, 4]-code or 39-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2184, 65575, F2, 23) (dual of [65575, 65391, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2184, 65572, F2, 23) (dual of [65572, 65388, 24]-code), using
- net defined by OOA [i] based on linear OOA(2184, 5961, F2, 23, 23) (dual of [(5961, 23), 136919, 24]-NRT-code), using
(164, 164+23, 11963)-Net over F2 — Digital
Digital (164, 187, 11963)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2187, 11963, F2, 5, 23) (dual of [(11963, 5), 59628, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2187, 13115, F2, 5, 23) (dual of [(13115, 5), 65388, 24]-NRT-code), using
- 23 times duplication [i] based on linear OOA(2184, 13115, F2, 5, 23) (dual of [(13115, 5), 65391, 24]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2184, 65575, F2, 23) (dual of [65575, 65391, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(2177, 65536, F2, 23) (dual of [65536, 65359, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2145, 65536, F2, 19) (dual of [65536, 65391, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(27, 39, F2, 3) (dual of [39, 32, 4]-code or 39-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- OOA 5-folding [i] based on linear OA(2184, 65575, F2, 23) (dual of [65575, 65391, 24]-code), using
- 23 times duplication [i] based on linear OOA(2184, 13115, F2, 5, 23) (dual of [(13115, 5), 65391, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2187, 13115, F2, 5, 23) (dual of [(13115, 5), 65388, 24]-NRT-code), using
(164, 164+23, 604152)-Net in Base 2 — Upper bound on s
There is no (164, 187, 604153)-net in base 2, because
- 1 times m-reduction [i] would yield (164, 186, 604153)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 98 081499 354755 926393 713604 494296 873426 490446 388519 168056 > 2186 [i]