Best Known (141, 165, s)-Nets in Base 2
(141, 165, 685)-Net over F2 — Constructive and digital
Digital (141, 165, 685)-net over F2, using
- 22 times duplication [i] based on digital (139, 163, 685)-net over F2, using
- t-expansion [i] based on digital (138, 163, 685)-net over F2, using
- net defined by OOA [i] based on linear OOA(2163, 685, F2, 25, 25) (dual of [(685, 25), 16962, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2163, 8221, F2, 25) (dual of [8221, 8058, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2163, 8225, F2, 25) (dual of [8225, 8062, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2157, 8193, F2, 25) (dual of [8193, 8036, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2131, 8193, F2, 21) (dual of [8193, 8062, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, 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(2163, 8225, F2, 25) (dual of [8225, 8062, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2163, 8221, F2, 25) (dual of [8221, 8058, 26]-code), using
- net defined by OOA [i] based on linear OOA(2163, 685, F2, 25, 25) (dual of [(685, 25), 16962, 26]-NRT-code), using
- t-expansion [i] based on digital (138, 163, 685)-net over F2, using
(141, 165, 2250)-Net over F2 — Digital
Digital (141, 165, 2250)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2165, 2250, F2, 3, 24) (dual of [(2250, 3), 6585, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2165, 2742, F2, 3, 24) (dual of [(2742, 3), 8061, 25]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2164, 2742, F2, 3, 24) (dual of [(2742, 3), 8062, 25]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2164, 8226, F2, 24) (dual of [8226, 8062, 25]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2162, 8224, F2, 24) (dual of [8224, 8062, 25]-code), using
- 1 times truncation [i] based on linear OA(2163, 8225, F2, 25) (dual of [8225, 8062, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2157, 8193, F2, 25) (dual of [8193, 8036, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2131, 8193, F2, 21) (dual of [8193, 8062, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, 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(2163, 8225, F2, 25) (dual of [8225, 8062, 26]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2162, 8224, F2, 24) (dual of [8224, 8062, 25]-code), using
- OOA 3-folding [i] based on linear OA(2164, 8226, F2, 24) (dual of [8226, 8062, 25]-code), using
- 21 times duplication [i] based on linear OOA(2164, 2742, F2, 3, 24) (dual of [(2742, 3), 8062, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2165, 2742, F2, 3, 24) (dual of [(2742, 3), 8061, 25]-NRT-code), using
(141, 165, 72848)-Net in Base 2 — Upper bound on s
There is no (141, 165, 72849)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 46 773278 803364 292357 441779 791299 125292 513750 903964 > 2165 [i]