Best Known (231−30, 231, s)-Nets in Base 2
(231−30, 231, 2186)-Net over F2 — Constructive and digital
Digital (201, 231, 2186)-net over F2, using
- net defined by OOA [i] based on linear OOA(2231, 2186, F2, 30, 30) (dual of [(2186, 30), 65349, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(2231, 32790, F2, 30) (dual of [32790, 32559, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2231, 32799, F2, 30) (dual of [32799, 32568, 31]-code), using
- 1 times truncation [i] 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
- 1 times truncation [i] based on linear OA(2232, 32800, F2, 31) (dual of [32800, 32568, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2231, 32799, F2, 30) (dual of [32799, 32568, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(2231, 32790, F2, 30) (dual of [32790, 32559, 31]-code), using
(231−30, 231, 6559)-Net over F2 — Digital
Digital (201, 231, 6559)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2231, 6559, F2, 5, 30) (dual of [(6559, 5), 32564, 31]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2231, 32795, F2, 30) (dual of [32795, 32564, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2231, 32799, F2, 30) (dual of [32799, 32568, 31]-code), using
- 1 times truncation [i] 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
- 1 times truncation [i] based on linear OA(2232, 32800, F2, 31) (dual of [32800, 32568, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2231, 32799, F2, 30) (dual of [32799, 32568, 31]-code), using
- OOA 5-folding [i] based on linear OA(2231, 32795, F2, 30) (dual of [32795, 32564, 31]-code), using
(231−30, 231, 277711)-Net in Base 2 — Upper bound on s
There is no (201, 231, 277712)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 3450 929648 750256 428550 644045 588859 622696 294628 696353 114047 089667 934503 > 2231 [i]