Best Known (220−29, 220, s)-Nets in Base 2
(220−29, 220, 2343)-Net over F2 — Constructive and digital
Digital (191, 220, 2343)-net over F2, using
- 22 times duplication [i] based on digital (189, 218, 2343)-net over F2, using
- net defined by OOA [i] based on linear OOA(2218, 2343, F2, 29, 29) (dual of [(2343, 29), 67729, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2218, 32803, F2, 29) (dual of [32803, 32585, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2218, 32806, F2, 29) (dual of [32806, 32588, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2211, 32769, F2, 29) (dual of [32769, 32558, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2181, 32769, F2, 25) (dual of [32769, 32588, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [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 C([0,14]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2218, 32806, F2, 29) (dual of [32806, 32588, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2218, 32803, F2, 29) (dual of [32803, 32585, 30]-code), using
- net defined by OOA [i] based on linear OOA(2218, 2343, F2, 29, 29) (dual of [(2343, 29), 67729, 30]-NRT-code), using
(220−29, 220, 6110)-Net over F2 — Digital
Digital (191, 220, 6110)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2220, 6110, F2, 5, 29) (dual of [(6110, 5), 30330, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2220, 6561, F2, 5, 29) (dual of [(6561, 5), 32585, 30]-NRT-code), using
- 22 times duplication [i] based on linear OOA(2218, 6561, F2, 5, 29) (dual of [(6561, 5), 32587, 30]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2218, 32805, F2, 29) (dual of [32805, 32587, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2218, 32806, F2, 29) (dual of [32806, 32588, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2211, 32769, F2, 29) (dual of [32769, 32558, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2181, 32769, F2, 25) (dual of [32769, 32588, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [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 C([0,14]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2218, 32806, F2, 29) (dual of [32806, 32588, 30]-code), using
- OOA 5-folding [i] based on linear OA(2218, 32805, F2, 29) (dual of [32805, 32587, 30]-code), using
- 22 times duplication [i] based on linear OOA(2218, 6561, F2, 5, 29) (dual of [(6561, 5), 32587, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2220, 6561, F2, 5, 29) (dual of [(6561, 5), 32585, 30]-NRT-code), using
(220−29, 220, 309313)-Net in Base 2 — Upper bound on s
There is no (191, 220, 309314)-net in base 2, because
- 1 times m-reduction [i] would yield (191, 219, 309314)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 842536 391412 212753 934492 556654 079192 682581 587312 983227 152437 065696 > 2219 [i]