Best Known (232−30, 232, s)-Nets in Base 2
(232−30, 232, 2186)-Net over F2 — Constructive and digital
Digital (202, 232, 2186)-net over F2, using
- t-expansion [i] based on digital (201, 232, 2186)-net over F2, using
- net defined by OOA [i] based on linear OOA(2232, 2186, F2, 31, 31) (dual of [(2186, 31), 67534, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2232, 32791, F2, 31) (dual of [32791, 32559, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2232, 32800, F2, 31) (dual of [32800, 32568, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- linear OA(2226, 32768, F2, 31) (dual of [32768, 32542, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 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(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(2232, 32800, F2, 31) (dual of [32800, 32568, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2232, 32791, F2, 31) (dual of [32791, 32559, 32]-code), using
- net defined by OOA [i] based on linear OOA(2232, 2186, F2, 31, 31) (dual of [(2186, 31), 67534, 32]-NRT-code), using
(232−30, 232, 6560)-Net over F2 — Digital
Digital (202, 232, 6560)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2232, 6560, F2, 5, 30) (dual of [(6560, 5), 32568, 31]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2232, 32800, F2, 30) (dual of [32800, 32568, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2232, 32804, F2, 30) (dual of [32804, 32572, 31]-code), using
- 1 times truncation [i] based on linear OA(2233, 32805, F2, 31) (dual of [32805, 32572, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- linear OA(2226, 32768, F2, 31) (dual of [32768, 32542, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 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(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(30) ⊂ Ce(26) [i] based on
- 1 times truncation [i] based on linear OA(2233, 32805, F2, 31) (dual of [32805, 32572, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2232, 32804, F2, 30) (dual of [32804, 32572, 31]-code), using
- OOA 5-folding [i] based on linear OA(2232, 32800, F2, 30) (dual of [32800, 32568, 31]-code), using
(232−30, 232, 290846)-Net in Base 2 — Upper bound on s
There is no (202, 232, 290847)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 6901 793154 491597 116971 801863 916269 381672 070165 818300 107862 387769 334346 > 2232 [i]