Best Known (232, 232+28, s)-Nets in Base 2
(232, 232+28, 18727)-Net over F2 — Constructive and digital
Digital (232, 260, 18727)-net over F2, using
- t-expansion [i] based on digital (231, 260, 18727)-net over F2, using
- net defined by OOA [i] based on linear OOA(2260, 18727, F2, 29, 29) (dual of [(18727, 29), 542823, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2260, 262179, F2, 29) (dual of [262179, 261919, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2260, 262188, F2, 29) (dual of [262188, 261928, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2253, 262145, F2, 29) (dual of [262145, 261892, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 236−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2217, 262145, F2, 25) (dual of [262145, 261928, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 236−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(27, 43, F2, 3) (dual of [43, 36, 4]-code or 43-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(2260, 262188, F2, 29) (dual of [262188, 261928, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2260, 262179, F2, 29) (dual of [262179, 261919, 30]-code), using
- net defined by OOA [i] based on linear OOA(2260, 18727, F2, 29, 29) (dual of [(18727, 29), 542823, 30]-NRT-code), using
(232, 232+28, 37947)-Net over F2 — Digital
Digital (232, 260, 37947)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2260, 37947, F2, 6, 28) (dual of [(37947, 6), 227422, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2260, 43698, F2, 6, 28) (dual of [(43698, 6), 261928, 29]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2260, 262188, F2, 28) (dual of [262188, 261928, 29]-code), using
- strength reduction [i] based on linear OA(2260, 262188, F2, 29) (dual of [262188, 261928, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2253, 262145, F2, 29) (dual of [262145, 261892, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 236−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2217, 262145, F2, 25) (dual of [262145, 261928, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 236−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(27, 43, F2, 3) (dual of [43, 36, 4]-code or 43-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
- strength reduction [i] based on linear OA(2260, 262188, F2, 29) (dual of [262188, 261928, 30]-code), using
- OOA 6-folding [i] based on linear OA(2260, 262188, F2, 28) (dual of [262188, 261928, 29]-code), using
- discarding factors / shortening the dual code based on linear OOA(2260, 43698, F2, 6, 28) (dual of [(43698, 6), 261928, 29]-NRT-code), using
(232, 232+28, 2355109)-Net in Base 2 — Upper bound on s
There is no (232, 260, 2355110)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1 852683 880493 262648 745984 110898 655631 326357 410577 886321 731671 863550 634504 768368 > 2260 [i]