Best Known (81, 81+12, s)-Nets in Base 2
(81, 81+12, 5464)-Net over F2 — Constructive and digital
Digital (81, 93, 5464)-net over F2, using
- 21 times duplication [i] based on digital (80, 92, 5464)-net over F2, using
- t-expansion [i] based on digital (79, 92, 5464)-net over F2, using
- net defined by OOA [i] based on linear OOA(292, 5464, F2, 13, 13) (dual of [(5464, 13), 70940, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(292, 32785, F2, 13) (dual of [32785, 32693, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(291, 32768, F2, 13) (dual of [32768, 32677, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(276, 32768, F2, 11) (dual of [32768, 32692, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(216, 17, F2, 15) (dual of [17, 1, 16]-code), using
- strength reduction [i] based on linear OA(216, 17, F2, 16) (dual of [17, 1, 17]-code or 17-arc in PG(15,2)), using
- dual of repetition code with length 17 [i]
- strength reduction [i] based on linear OA(216, 17, F2, 16) (dual of [17, 1, 17]-code or 17-arc in PG(15,2)), using
- 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 X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(292, 32785, F2, 13) (dual of [32785, 32693, 14]-code), using
- net defined by OOA [i] based on linear OOA(292, 5464, F2, 13, 13) (dual of [(5464, 13), 70940, 14]-NRT-code), using
- t-expansion [i] based on digital (79, 92, 5464)-net over F2, using
(81, 81+12, 9157)-Net over F2 — Digital
Digital (81, 93, 9157)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(293, 9157, F2, 3, 12) (dual of [(9157, 3), 27378, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(293, 10928, F2, 3, 12) (dual of [(10928, 3), 32691, 13]-NRT-code), using
- 1 step truncation [i] based on linear OOA(294, 10929, F2, 3, 13) (dual of [(10929, 3), 32693, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(294, 32787, F2, 13) (dual of [32787, 32693, 14]-code), using
- 2 times code embedding in larger space [i] based on linear OA(292, 32785, F2, 13) (dual of [32785, 32693, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(291, 32768, F2, 13) (dual of [32768, 32677, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(276, 32768, F2, 11) (dual of [32768, 32692, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(216, 17, F2, 15) (dual of [17, 1, 16]-code), using
- strength reduction [i] based on linear OA(216, 17, F2, 16) (dual of [17, 1, 17]-code or 17-arc in PG(15,2)), using
- dual of repetition code with length 17 [i]
- strength reduction [i] based on linear OA(216, 17, F2, 16) (dual of [17, 1, 17]-code or 17-arc in PG(15,2)), using
- 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 X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(292, 32785, F2, 13) (dual of [32785, 32693, 14]-code), using
- OOA 3-folding [i] based on linear OA(294, 32787, F2, 13) (dual of [32787, 32693, 14]-code), using
- 1 step truncation [i] based on linear OOA(294, 10929, F2, 3, 13) (dual of [(10929, 3), 32693, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(293, 10928, F2, 3, 12) (dual of [(10928, 3), 32691, 13]-NRT-code), using
(81, 81+12, 138726)-Net in Base 2 — Upper bound on s
There is no (81, 93, 138727)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 9903 600691 263245 034424 942741 > 293 [i]