Best Known (117, 117+15, s)-Nets in Base 2
(117, 117+15, 37452)-Net over F2 — Constructive and digital
Digital (117, 132, 37452)-net over F2, using
- 23 times duplication [i] based on digital (114, 129, 37452)-net over F2, using
- net defined by OOA [i] based on linear OOA(2129, 37452, F2, 15, 15) (dual of [(37452, 15), 561651, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2129, 262165, F2, 15) (dual of [262165, 262036, 16]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2128, 262164, F2, 15) (dual of [262164, 262036, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(2127, 262144, F2, 15) (dual of [262144, 262017, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2109, 262144, F2, 13) (dual of [262144, 262035, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(219, 20, F2, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,2)), using
- dual of repetition code with length 20 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2128, 262164, F2, 15) (dual of [262164, 262036, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2129, 262165, F2, 15) (dual of [262165, 262036, 16]-code), using
- net defined by OOA [i] based on linear OOA(2129, 37452, F2, 15, 15) (dual of [(37452, 15), 561651, 16]-NRT-code), using
(117, 117+15, 52433)-Net over F2 — Digital
Digital (117, 132, 52433)-net over F2, using
- 23 times duplication [i] based on digital (114, 129, 52433)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2129, 52433, F2, 5, 15) (dual of [(52433, 5), 262036, 16]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2129, 262165, F2, 15) (dual of [262165, 262036, 16]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2128, 262164, F2, 15) (dual of [262164, 262036, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(2127, 262144, F2, 15) (dual of [262144, 262017, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2109, 262144, F2, 13) (dual of [262144, 262035, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(219, 20, F2, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,2)), using
- dual of repetition code with length 20 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2128, 262164, F2, 15) (dual of [262164, 262036, 16]-code), using
- OOA 5-folding [i] based on linear OA(2129, 262165, F2, 15) (dual of [262165, 262036, 16]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2129, 52433, F2, 5, 15) (dual of [(52433, 5), 262036, 16]-NRT-code), using
(117, 117+15, 1453707)-Net in Base 2 — Upper bound on s
There is no (117, 132, 1453708)-net in base 2, because
- 1 times m-reduction [i] would yield (117, 131, 1453708)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 2722 266501 026626 511105 847117 925801 468934 > 2131 [i]