Best Known (89, 89+12, s)-Nets in Base 2
(89, 89+12, 10926)-Net over F2 — Constructive and digital
Digital (89, 101, 10926)-net over F2, using
- t-expansion [i] based on digital (88, 101, 10926)-net over F2, using
- net defined by OOA [i] based on linear OOA(2101, 10926, F2, 13, 13) (dual of [(10926, 13), 141937, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2101, 65557, F2, 13) (dual of [65557, 65456, 14]-code), using
- 3 times code embedding in larger space [i] based on linear OA(298, 65554, F2, 13) (dual of [65554, 65456, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- 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(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(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(12) ⊂ Ce(10) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(298, 65554, F2, 13) (dual of [65554, 65456, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2101, 65557, F2, 13) (dual of [65557, 65456, 14]-code), using
- net defined by OOA [i] based on linear OOA(2101, 10926, F2, 13, 13) (dual of [(10926, 13), 141937, 14]-NRT-code), using
(89, 89+12, 18325)-Net over F2 — Digital
Digital (89, 101, 18325)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2101, 18325, F2, 3, 12) (dual of [(18325, 3), 54874, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2101, 21852, F2, 3, 12) (dual of [(21852, 3), 65455, 13]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2100, 21852, F2, 3, 12) (dual of [(21852, 3), 65456, 13]-NRT-code), using
- strength reduction [i] based on linear OOA(2100, 21852, F2, 3, 13) (dual of [(21852, 3), 65456, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2100, 65556, F2, 13) (dual of [65556, 65456, 14]-code), using
- 2 times code embedding in larger space [i] based on linear OA(298, 65554, F2, 13) (dual of [65554, 65456, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- 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(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(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(12) ⊂ Ce(10) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(298, 65554, F2, 13) (dual of [65554, 65456, 14]-code), using
- OOA 3-folding [i] based on linear OA(2100, 65556, F2, 13) (dual of [65556, 65456, 14]-code), using
- strength reduction [i] based on linear OOA(2100, 21852, F2, 3, 13) (dual of [(21852, 3), 65456, 14]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2100, 21852, F2, 3, 12) (dual of [(21852, 3), 65456, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2101, 21852, F2, 3, 12) (dual of [(21852, 3), 65455, 13]-NRT-code), using
(89, 89+12, 349582)-Net in Base 2 — Upper bound on s
There is no (89, 101, 349583)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2 535319 464050 689859 910740 206367 > 2101 [i]