Best Known (218−20, 218, s)-Nets in Base 2
(218−20, 218, 209720)-Net over F2 — Constructive and digital
Digital (198, 218, 209720)-net over F2, using
- t-expansion [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
(218−20, 218, 349533)-Net over F2 — Digital
Digital (198, 218, 349533)-net over F2, using
- 21 times duplication [i] based on digital (197, 217, 349533)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2217, 349533, F2, 6, 20) (dual of [(349533, 6), 2096981, 21]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2217, 2097198, F2, 20) (dual of [2097198, 2096981, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(2217, 2097201, F2, 20) (dual of [2097201, 2096984, 21]-code), using
- 1 times truncation [i] 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
- 1 times truncation [i] based on linear OA(2218, 2097202, F2, 21) (dual of [2097202, 2096984, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2217, 2097201, F2, 20) (dual of [2097201, 2096984, 21]-code), using
- OOA 6-folding [i] based on linear OA(2217, 2097198, F2, 20) (dual of [2097198, 2096981, 21]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2217, 349533, F2, 6, 20) (dual of [(349533, 6), 2096981, 21]-NRT-code), using
(218−20, 218, large)-Net in Base 2 — Upper bound on s
There is no (198, 218, large)-net in base 2, because
- 18 times m-reduction [i] would yield (198, 200, large)-net in base 2, but