Best Known (99−16, 99, s)-Nets in Base 3
(99−16, 99, 2464)-Net over F3 — Constructive and digital
Digital (83, 99, 2464)-net over F3, using
- net defined by OOA [i] based on linear OOA(399, 2464, F3, 16, 16) (dual of [(2464, 16), 39325, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(399, 19712, F3, 16) (dual of [19712, 19613, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(399, 19718, F3, 16) (dual of [19718, 19619, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(364, 19683, F3, 11) (dual of [19683, 19619, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(38, 35, F3, 4) (dual of [35, 27, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(399, 19718, F3, 16) (dual of [19718, 19619, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(399, 19712, F3, 16) (dual of [19712, 19613, 17]-code), using
(99−16, 99, 9859)-Net over F3 — Digital
Digital (83, 99, 9859)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(399, 9859, F3, 2, 16) (dual of [(9859, 2), 19619, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(399, 19718, F3, 16) (dual of [19718, 19619, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(364, 19683, F3, 11) (dual of [19683, 19619, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(38, 35, F3, 4) (dual of [35, 27, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- OOA 2-folding [i] based on linear OA(399, 19718, F3, 16) (dual of [19718, 19619, 17]-code), using
(99−16, 99, 1510196)-Net in Base 3 — Upper bound on s
There is no (83, 99, 1510197)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 171793 023008 267050 681908 052782 106295 983159 418721 > 399 [i]