Best Known (133−17, 133, s)-Nets in Base 2
(133−17, 133, 8194)-Net over F2 — Constructive and digital
Digital (116, 133, 8194)-net over F2, using
- 23 times duplication [i] based on digital (113, 130, 8194)-net over F2, using
- net defined by OOA [i] based on linear OOA(2130, 8194, F2, 17, 17) (dual of [(8194, 17), 139168, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2130, 65553, F2, 17) (dual of [65553, 65423, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2129, 65536, F2, 17) (dual of [65536, 65407, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2113, 65536, F2, 15) (dual of [65536, 65423, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(21, 17, F2, 1) (dual of [17, 16, 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 X applied to Ce(16) ⊂ Ce(14) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(2130, 65553, F2, 17) (dual of [65553, 65423, 18]-code), using
- net defined by OOA [i] based on linear OOA(2130, 8194, F2, 17, 17) (dual of [(8194, 17), 139168, 18]-NRT-code), using
(133−17, 133, 13111)-Net over F2 — Digital
Digital (116, 133, 13111)-net over F2, using
- 22 times duplication [i] based on digital (114, 131, 13111)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2131, 13111, F2, 5, 17) (dual of [(13111, 5), 65424, 18]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2131, 65555, F2, 17) (dual of [65555, 65424, 18]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2130, 65554, F2, 17) (dual of [65554, 65424, 18]-code), using
- construction X4 applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2129, 65536, F2, 17) (dual of [65536, 65407, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2113, 65536, F2, 15) (dual of [65536, 65423, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(217, 18, F2, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,2)), using
- dual of repetition code with length 18 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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(16) ⊂ Ce(14) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2130, 65554, F2, 17) (dual of [65554, 65424, 18]-code), using
- OOA 5-folding [i] based on linear OA(2131, 65555, F2, 17) (dual of [65555, 65424, 18]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2131, 13111, F2, 5, 17) (dual of [(13111, 5), 65424, 18]-NRT-code), using
(133−17, 133, 348875)-Net in Base 2 — Upper bound on s
There is no (116, 133, 348876)-net in base 2, because
- 1 times m-reduction [i] would yield (116, 132, 348876)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 5444 559557 812582 741249 402070 823191 189581 > 2132 [i]