Best Known (199, 220, s)-Nets in Base 2
(199, 220, 209720)-Net over F2 — Constructive and digital
Digital (199, 220, 209720)-net over F2, using
- 22 times duplication [i] based on digital (197, 218, 209720)-net over F2, using
- net defined by OOA [i] based on linear OOA(2218, 209720, F2, 21, 21) (dual of [(209720, 21), 4403902, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2218, 2097201, F2, 21) (dual of [2097201, 2096983, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2218, 2097202, F2, 21) (dual of [2097202, 2096984, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2211, 2097153, F2, 21) (dual of [2097153, 2096942, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 242−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2169, 2097153, F2, 17) (dual of [2097153, 2096984, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 242−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(27, 49, F2, 3) (dual of [49, 42, 4]-code or 49-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([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2218, 2097202, F2, 21) (dual of [2097202, 2096984, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2218, 2097201, F2, 21) (dual of [2097201, 2096983, 22]-code), using
- net defined by OOA [i] based on linear OOA(2218, 209720, F2, 21, 21) (dual of [(209720, 21), 4403902, 22]-NRT-code), using
(199, 220, 299600)-Net over F2 — Digital
Digital (199, 220, 299600)-net over F2, using
- 22 times duplication [i] based on digital (197, 218, 299600)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2218, 299600, F2, 7, 21) (dual of [(299600, 7), 2096982, 22]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2218, 2097200, F2, 21) (dual of [2097200, 2096982, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2218, 2097202, F2, 21) (dual of [2097202, 2096984, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2211, 2097153, F2, 21) (dual of [2097153, 2096942, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 242−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2169, 2097153, F2, 17) (dual of [2097153, 2096984, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 242−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(27, 49, F2, 3) (dual of [49, 42, 4]-code or 49-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([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2218, 2097202, F2, 21) (dual of [2097202, 2096984, 22]-code), using
- OOA 7-folding [i] based on linear OA(2218, 2097200, F2, 21) (dual of [2097200, 2096982, 22]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2218, 299600, F2, 7, 21) (dual of [(299600, 7), 2096982, 22]-NRT-code), using
(199, 220, large)-Net in Base 2 — Upper bound on s
There is no (199, 220, large)-net in base 2, because
- 19 times m-reduction [i] would yield (199, 201, large)-net in base 2, but