Best Known (176, 176+27, s)-Nets in Base 2
(176, 176+27, 2523)-Net over F2 — Constructive and digital
Digital (176, 203, 2523)-net over F2, using
- 21 times duplication [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
(176, 176+27, 5951)-Net over F2 — Digital
Digital (176, 203, 5951)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2203, 5951, F2, 5, 27) (dual of [(5951, 5), 29552, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2203, 6561, F2, 5, 27) (dual of [(6561, 5), 32602, 28]-NRT-code), using
- OOA 5-folding [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
- OOA 5-folding [i] based on linear OA(2203, 32805, F2, 27) (dual of [32805, 32602, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(2203, 6561, F2, 5, 27) (dual of [(6561, 5), 32602, 28]-NRT-code), using
(176, 176+27, 269723)-Net in Base 2 — Upper bound on s
There is no (176, 203, 269724)-net in base 2, because
- 1 times m-reduction [i] would yield (176, 202, 269724)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 6 427943 543299 182324 804614 820505 046808 095607 513532 991926 508588 > 2202 [i]