Best Known (259−25, 259, s)-Nets in Base 2
(259−25, 259, 174765)-Net over F2 — Constructive and digital
Digital (234, 259, 174765)-net over F2, using
- net defined by OOA [i] based on linear OOA(2259, 174765, F2, 25, 25) (dual of [(174765, 25), 4368866, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2259, 2097181, F2, 25) (dual of [2097181, 2096922, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2259, 2097185, F2, 25) (dual of [2097185, 2096926, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2253, 2097153, F2, 25) (dual of [2097153, 2096900, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 242−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 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(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2259, 2097185, F2, 25) (dual of [2097185, 2096926, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2259, 2097181, F2, 25) (dual of [2097181, 2096922, 26]-code), using
(259−25, 259, 262148)-Net over F2 — Digital
Digital (234, 259, 262148)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2259, 262148, F2, 8, 25) (dual of [(262148, 8), 2096925, 26]-NRT-code), using
- OOA 8-folding [i] based on linear OA(2259, 2097184, F2, 25) (dual of [2097184, 2096925, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2259, 2097185, F2, 25) (dual of [2097185, 2096926, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2253, 2097153, F2, 25) (dual of [2097153, 2096900, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 242−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 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(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2259, 2097185, F2, 25) (dual of [2097185, 2096926, 26]-code), using
- OOA 8-folding [i] based on linear OA(2259, 2097184, F2, 25) (dual of [2097184, 2096925, 26]-code), using
(259−25, 259, large)-Net in Base 2 — Upper bound on s
There is no (234, 259, large)-net in base 2, because
- 23 times m-reduction [i] would yield (234, 236, large)-net in base 2, but