Best Known (116−15, 116, s)-Nets in Base 2
(116−15, 116, 9365)-Net over F2 — Constructive and digital
Digital (101, 116, 9365)-net over F2, using
- net defined by OOA [i] based on linear OOA(2116, 9365, F2, 15, 15) (dual of [(9365, 15), 140359, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2116, 65556, F2, 15) (dual of [65556, 65440, 16]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2114, 65554, F2, 15) (dual of [65554, 65440, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- 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(297, 65536, F2, 13) (dual of [65536, 65439, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(14) ⊂ Ce(12) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2114, 65554, F2, 15) (dual of [65554, 65440, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2116, 65556, F2, 15) (dual of [65556, 65440, 16]-code), using
(116−15, 116, 13111)-Net over F2 — Digital
Digital (101, 116, 13111)-net over F2, using
- 21 times duplication [i] based on digital (100, 115, 13111)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2115, 13111, F2, 5, 15) (dual of [(13111, 5), 65440, 16]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2115, 65555, F2, 15) (dual of [65555, 65440, 16]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2114, 65554, F2, 15) (dual of [65554, 65440, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- 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(297, 65536, F2, 13) (dual of [65536, 65439, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(14) ⊂ Ce(12) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2114, 65554, F2, 15) (dual of [65554, 65440, 16]-code), using
- OOA 5-folding [i] based on linear OA(2115, 65555, F2, 15) (dual of [65555, 65440, 16]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2115, 13111, F2, 5, 15) (dual of [(13111, 5), 65440, 16]-NRT-code), using
(116−15, 116, 298123)-Net in Base 2 — Upper bound on s
There is no (101, 116, 298124)-net in base 2, because
- 1 times m-reduction [i] would yield (101, 115, 298124)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 41538 423593 401141 312817 816930 383494 > 2115 [i]