Best Known (234, 258, s)-Nets in Base 2
(234, 258, 174765)-Net over F2 — Constructive and digital
Digital (234, 258, 174765)-net over F2, using
- net defined by OOA [i] based on linear OOA(2258, 174765, F2, 24, 24) (dual of [(174765, 24), 4194102, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(2258, 2097180, F2, 24) (dual of [2097180, 2096922, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(2258, 2097184, F2, 24) (dual of [2097184, 2096926, 25]-code), using
- 1 times truncation [i] 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
- 1 times truncation [i] based on linear OA(2259, 2097185, F2, 25) (dual of [2097185, 2096926, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2258, 2097184, F2, 24) (dual of [2097184, 2096926, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(2258, 2097180, F2, 24) (dual of [2097180, 2096922, 25]-code), using
(234, 258, 299597)-Net over F2 — Digital
Digital (234, 258, 299597)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2258, 299597, F2, 7, 24) (dual of [(299597, 7), 2096921, 25]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2258, 2097179, F2, 24) (dual of [2097179, 2096921, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(2258, 2097184, F2, 24) (dual of [2097184, 2096926, 25]-code), using
- 1 times truncation [i] 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
- 1 times truncation [i] based on linear OA(2259, 2097185, F2, 25) (dual of [2097185, 2096926, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2258, 2097184, F2, 24) (dual of [2097184, 2096926, 25]-code), using
- OOA 7-folding [i] based on linear OA(2258, 2097179, F2, 24) (dual of [2097179, 2096921, 25]-code), using
(234, 258, large)-Net in Base 2 — Upper bound on s
There is no (234, 258, large)-net in base 2, because
- 22 times m-reduction [i] would yield (234, 236, large)-net in base 2, but