Best Known (219, 219+29, s)-Nets in Base 2
(219, 219+29, 9365)-Net over F2 — Constructive and digital
Digital (219, 248, 9365)-net over F2, using
- 22 times duplication [i] based on digital (217, 246, 9365)-net over F2, using
- net defined by OOA [i] based on linear OOA(2246, 9365, F2, 29, 29) (dual of [(9365, 29), 271339, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2246, 131111, F2, 29) (dual of [131111, 130865, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2246, 131114, F2, 29) (dual of [131114, 130868, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2239, 131073, F2, 29) (dual of [131073, 130834, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 131073 | 234−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2205, 131073, F2, 25) (dual of [131073, 130868, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 131073 | 234−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(27, 41, F2, 3) (dual of [41, 34, 4]-code or 41-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(2246, 131114, F2, 29) (dual of [131114, 130868, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2246, 131111, F2, 29) (dual of [131111, 130865, 30]-code), using
- net defined by OOA [i] based on linear OOA(2246, 9365, F2, 29, 29) (dual of [(9365, 29), 271339, 30]-NRT-code), using
(219, 219+29, 18730)-Net over F2 — Digital
Digital (219, 248, 18730)-net over F2, using
- 22 times duplication [i] based on digital (217, 246, 18730)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2246, 18730, F2, 7, 29) (dual of [(18730, 7), 130864, 30]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2246, 131110, F2, 29) (dual of [131110, 130864, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2246, 131114, F2, 29) (dual of [131114, 130868, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2239, 131073, F2, 29) (dual of [131073, 130834, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 131073 | 234−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2205, 131073, F2, 25) (dual of [131073, 130868, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 131073 | 234−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(27, 41, F2, 3) (dual of [41, 34, 4]-code or 41-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(2246, 131114, F2, 29) (dual of [131114, 130868, 30]-code), using
- OOA 7-folding [i] based on linear OA(2246, 131110, F2, 29) (dual of [131110, 130864, 30]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2246, 18730, F2, 7, 29) (dual of [(18730, 7), 130864, 30]-NRT-code), using
(219, 219+29, 1237313)-Net in Base 2 — Upper bound on s
There is no (219, 248, 1237314)-net in base 2, because
- 1 times m-reduction [i] would yield (219, 247, 1237314)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 226 157690 029978 004214 609594 352598 394898 374959 850273 385775 541756 287574 931296 > 2247 [i]