Best Known (227−20, 227, s)-Nets in Base 2
(227−20, 227, 419435)-Net over F2 — Constructive and digital
Digital (207, 227, 419435)-net over F2, using
- net defined by OOA [i] based on linear OOA(2227, 419435, F2, 20, 20) (dual of [(419435, 20), 8388473, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(2227, 4194350, F2, 20) (dual of [4194350, 4194123, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(2227, 4194355, F2, 20) (dual of [4194355, 4194128, 21]-code), using
- 1 times truncation [i] based on linear OA(2228, 4194356, F2, 21) (dual of [4194356, 4194128, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2221, 4194305, F2, 21) (dual of [4194305, 4194084, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 244−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2177, 4194305, F2, 17) (dual of [4194305, 4194128, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 244−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(27, 51, F2, 3) (dual of [51, 44, 4]-code or 51-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(2228, 4194356, F2, 21) (dual of [4194356, 4194128, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2227, 4194355, F2, 20) (dual of [4194355, 4194128, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(2227, 4194350, F2, 20) (dual of [4194350, 4194123, 21]-code), using
(227−20, 227, 699059)-Net over F2 — Digital
Digital (207, 227, 699059)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2227, 699059, F2, 6, 20) (dual of [(699059, 6), 4194127, 21]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2227, 4194354, F2, 20) (dual of [4194354, 4194127, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(2227, 4194355, F2, 20) (dual of [4194355, 4194128, 21]-code), using
- 1 times truncation [i] based on linear OA(2228, 4194356, F2, 21) (dual of [4194356, 4194128, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2221, 4194305, F2, 21) (dual of [4194305, 4194084, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 244−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2177, 4194305, F2, 17) (dual of [4194305, 4194128, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 244−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(27, 51, F2, 3) (dual of [51, 44, 4]-code or 51-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(2228, 4194356, F2, 21) (dual of [4194356, 4194128, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2227, 4194355, F2, 20) (dual of [4194355, 4194128, 21]-code), using
- OOA 6-folding [i] based on linear OA(2227, 4194354, F2, 20) (dual of [4194354, 4194127, 21]-code), using
(227−20, 227, large)-Net in Base 2 — Upper bound on s
There is no (207, 227, large)-net in base 2, because
- 18 times m-reduction [i] would yield (207, 209, large)-net in base 2, but