Best Known (251−24, 251, s)-Nets in Base 2
(251−24, 251, 87385)-Net over F2 — Constructive and digital
Digital (227, 251, 87385)-net over F2, using
- 23 times duplication [i] based on digital (224, 248, 87385)-net over F2, using
- t-expansion [i] based on digital (223, 248, 87385)-net over F2, using
- net defined by OOA [i] based on linear OOA(2248, 87385, F2, 25, 25) (dual of [(87385, 25), 2184377, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2248, 1048621, F2, 25) (dual of [1048621, 1048373, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2248, 1048624, F2, 25) (dual of [1048624, 1048376, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2241, 1048577, F2, 25) (dual of [1048577, 1048336, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2201, 1048577, F2, 21) (dual of [1048577, 1048376, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(27, 47, F2, 3) (dual of [47, 40, 4]-code or 47-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 C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2248, 1048624, F2, 25) (dual of [1048624, 1048376, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2248, 1048621, F2, 25) (dual of [1048621, 1048373, 26]-code), using
- net defined by OOA [i] based on linear OOA(2248, 87385, F2, 25, 25) (dual of [(87385, 25), 2184377, 26]-NRT-code), using
- t-expansion [i] based on digital (223, 248, 87385)-net over F2, using
(251−24, 251, 156405)-Net over F2 — Digital
Digital (227, 251, 156405)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2251, 156405, F2, 6, 24) (dual of [(156405, 6), 938179, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2251, 174771, F2, 6, 24) (dual of [(174771, 6), 1048375, 25]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2250, 174771, F2, 6, 24) (dual of [(174771, 6), 1048376, 25]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2250, 1048626, F2, 24) (dual of [1048626, 1048376, 25]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2247, 1048623, F2, 24) (dual of [1048623, 1048376, 25]-code), using
- 1 times truncation [i] based on linear OA(2248, 1048624, F2, 25) (dual of [1048624, 1048376, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2241, 1048577, F2, 25) (dual of [1048577, 1048336, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2201, 1048577, F2, 21) (dual of [1048577, 1048376, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(27, 47, F2, 3) (dual of [47, 40, 4]-code or 47-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 C([0,12]) ⊂ C([0,10]) [i] based on
- 1 times truncation [i] based on linear OA(2248, 1048624, F2, 25) (dual of [1048624, 1048376, 26]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2247, 1048623, F2, 24) (dual of [1048623, 1048376, 25]-code), using
- OOA 6-folding [i] based on linear OA(2250, 1048626, F2, 24) (dual of [1048626, 1048376, 25]-code), using
- 21 times duplication [i] based on linear OOA(2250, 174771, F2, 6, 24) (dual of [(174771, 6), 1048376, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2251, 174771, F2, 6, 24) (dual of [(174771, 6), 1048375, 25]-NRT-code), using
(251−24, 251, large)-Net in Base 2 — Upper bound on s
There is no (227, 251, large)-net in base 2, because
- 22 times m-reduction [i] would yield (227, 229, large)-net in base 2, but