Best Known (180, 206, s)-Nets in Base 2
(180, 206, 2523)-Net over F2 — Constructive and digital
Digital (180, 206, 2523)-net over F2, using
- 24 times duplication [i] based on digital (176, 202, 2523)-net over F2, using
- t-expansion [i] based on digital (175, 202, 2523)-net over F2, using
- net defined by OOA [i] based on linear OOA(2202, 2523, F2, 27, 27) (dual of [(2523, 27), 67919, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2202, 32800, F2, 27) (dual of [32800, 32598, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(2196, 32768, F2, 27) (dual of [32768, 32572, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2166, 32768, F2, 23) (dual of [32768, 32602, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- OOA 13-folding and stacking with additional row [i] based on linear OA(2202, 32800, F2, 27) (dual of [32800, 32598, 28]-code), using
- net defined by OOA [i] based on linear OOA(2202, 2523, F2, 27, 27) (dual of [(2523, 27), 67919, 28]-NRT-code), using
- t-expansion [i] based on digital (175, 202, 2523)-net over F2, using
(180, 206, 6795)-Net over F2 — Digital
Digital (180, 206, 6795)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2206, 6795, F2, 4, 26) (dual of [(6795, 4), 26974, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2206, 8202, F2, 4, 26) (dual of [(8202, 4), 32602, 27]-NRT-code), using
- strength reduction [i] based on linear OOA(2206, 8202, F2, 4, 27) (dual of [(8202, 4), 32602, 28]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2206, 32808, F2, 27) (dual of [32808, 32602, 28]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2203, 32805, F2, 27) (dual of [32805, 32602, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(2196, 32768, F2, 27) (dual of [32768, 32572, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2166, 32768, F2, 23) (dual of [32768, 32602, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(27, 37, F2, 3) (dual of [37, 30, 4]-code or 37-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(26) ⊂ Ce(22) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(2203, 32805, F2, 27) (dual of [32805, 32602, 28]-code), using
- OOA 4-folding [i] based on linear OA(2206, 32808, F2, 27) (dual of [32808, 32602, 28]-code), using
- strength reduction [i] based on linear OOA(2206, 8202, F2, 4, 27) (dual of [(8202, 4), 32602, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2206, 8202, F2, 4, 26) (dual of [(8202, 4), 32602, 27]-NRT-code), using
(180, 206, 333848)-Net in Base 2 — Upper bound on s
There is no (180, 206, 333849)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 102 847092 217868 210230 130245 512952 582681 168236 039962 683997 758088 > 2206 [i]