Best Known (205, 228, s)-Nets in Base 2
(205, 228, 95329)-Net over F2 — Constructive and digital
Digital (205, 228, 95329)-net over F2, using
- net defined by OOA [i] based on linear OOA(2228, 95329, F2, 23, 23) (dual of [(95329, 23), 2192339, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2228, 1048620, F2, 23) (dual of [1048620, 1048392, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2228, 1048623, F2, 23) (dual of [1048623, 1048395, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(2221, 1048576, F2, 23) (dual of [1048576, 1048355, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2181, 1048576, F2, 19) (dual of [1048576, 1048395, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(27, 47, F2, 3) (dual of [47, 40, 4]-code or 47-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 Ce(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2228, 1048623, F2, 23) (dual of [1048623, 1048395, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2228, 1048620, F2, 23) (dual of [1048620, 1048392, 24]-code), using
(205, 228, 149803)-Net over F2 — Digital
Digital (205, 228, 149803)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2228, 149803, F2, 7, 23) (dual of [(149803, 7), 1048393, 24]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2228, 1048621, F2, 23) (dual of [1048621, 1048393, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2228, 1048623, F2, 23) (dual of [1048623, 1048395, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(2221, 1048576, F2, 23) (dual of [1048576, 1048355, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2181, 1048576, F2, 19) (dual of [1048576, 1048395, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(27, 47, F2, 3) (dual of [47, 40, 4]-code or 47-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 Ce(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2228, 1048623, F2, 23) (dual of [1048623, 1048395, 24]-code), using
- OOA 7-folding [i] based on linear OA(2228, 1048621, F2, 23) (dual of [1048621, 1048393, 24]-code), using
(205, 228, 8001616)-Net in Base 2 — Upper bound on s
There is no (205, 228, 8001617)-net in base 2, because
- 1 times m-reduction [i] would yield (205, 227, 8001617)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 215 679661 538646 410348 417963 574006 692754 341892 443134 335027 166463 701552 > 2227 [i]