Best Known (85−11, 85, s)-Nets in Base 2
(85−11, 85, 13111)-Net over F2 — Constructive and digital
Digital (74, 85, 13111)-net over F2, using
- 21 times duplication [i] based on digital (73, 84, 13111)-net over F2, using
- net defined by OOA [i] based on linear OOA(284, 13111, F2, 11, 11) (dual of [(13111, 11), 144137, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(284, 65556, F2, 11) (dual of [65556, 65472, 12]-code), using
- 2 times code embedding in larger space [i] based on linear OA(282, 65554, F2, 11) (dual of [65554, 65472, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(281, 65536, F2, 11) (dual of [65536, 65455, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(265, 65536, F2, 9) (dual of [65536, 65471, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(10) ⊂ Ce(8) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(282, 65554, F2, 11) (dual of [65554, 65472, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(284, 65556, F2, 11) (dual of [65556, 65472, 12]-code), using
- net defined by OOA [i] based on linear OOA(284, 13111, F2, 11, 11) (dual of [(13111, 11), 144137, 12]-NRT-code), using
(85−11, 85, 16389)-Net over F2 — Digital
Digital (74, 85, 16389)-net over F2, using
- 21 times duplication [i] based on digital (73, 84, 16389)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(284, 16389, F2, 4, 11) (dual of [(16389, 4), 65472, 12]-NRT-code), using
- OOA 4-folding [i] based on linear OA(284, 65556, F2, 11) (dual of [65556, 65472, 12]-code), using
- 2 times code embedding in larger space [i] based on linear OA(282, 65554, F2, 11) (dual of [65554, 65472, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(281, 65536, F2, 11) (dual of [65536, 65455, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(265, 65536, F2, 9) (dual of [65536, 65471, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(10) ⊂ Ce(8) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(282, 65554, F2, 11) (dual of [65554, 65472, 12]-code), using
- OOA 4-folding [i] based on linear OA(284, 65556, F2, 11) (dual of [65556, 65472, 12]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(284, 16389, F2, 4, 11) (dual of [(16389, 4), 65472, 12]-NRT-code), using
(85−11, 85, 297255)-Net in Base 2 — Upper bound on s
There is no (74, 85, 297256)-net in base 2, because
- 1 times m-reduction [i] would yield (74, 84, 297256)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 19 342964 532484 521419 684883 > 284 [i]