Best Known (84, 84+16, s)-Nets in Base 3
(84, 84+16, 2467)-Net over F3 — Constructive and digital
Digital (84, 100, 2467)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (75, 91, 2460)-net over F3, using
- net defined by OOA [i] based on linear OOA(391, 2460, F3, 16, 16) (dual of [(2460, 16), 39269, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(391, 19680, F3, 16) (dual of [19680, 19589, 17]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(391, 19680, F3, 16) (dual of [19680, 19589, 17]-code), using
- net defined by OOA [i] based on linear OOA(391, 2460, F3, 16, 16) (dual of [(2460, 16), 39269, 17]-NRT-code), using
- digital (1, 9, 7)-net over F3, using
(84, 84+16, 9859)-Net over F3 — Digital
Digital (84, 100, 9859)-net over F3, using
- 31 times duplication [i] based on 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
- 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
(84, 84+16, 1732502)-Net in Base 3 — Upper bound on s
There is no (84, 100, 1732503)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 515378 675143 647887 026649 097604 804800 005169 326193 > 3100 [i]