Best Known (84, 84+15, s)-Nets in Base 3
(84, 84+15, 2820)-Net over F3 — Constructive and digital
Digital (84, 99, 2820)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 9, 9)-net over F3, using
- digital (75, 90, 2811)-net over F3, using
- net defined by OOA [i] based on linear OOA(390, 2811, F3, 15, 15) (dual of [(2811, 15), 42075, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(390, 19678, F3, 15) (dual of [19678, 19588, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(390, 19682, F3, 15) (dual of [19682, 19592, 16]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(390, 19682, F3, 15) (dual of [19682, 19592, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(390, 19678, F3, 15) (dual of [19678, 19588, 16]-code), using
- net defined by OOA [i] based on linear OOA(390, 2811, F3, 15, 15) (dual of [(2811, 15), 42075, 16]-NRT-code), using
(84, 84+15, 11186)-Net over F3 — Digital
Digital (84, 99, 11186)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(399, 11186, F3, 15) (dual of [11186, 11087, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(399, 19694, F3, 15) (dual of [19694, 19595, 16]-code), using
- (u, u+v)-construction [i] based on
- linear OA(39, 12, F3, 7) (dual of [12, 3, 8]-code), using
- 1 times truncation [i] based on linear OA(310, 13, F3, 8) (dual of [13, 3, 9]-code), using
- Simplex code S(3,3) [i]
- the expurgated narrow-sense BCH-code C(I) with length 13 | 33−1, defining interval I = [0,6], and minimum distance d ≥ |{−1,0,…,6}|+1 = 9 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(310, 13, F3, 8) (dual of [13, 3, 9]-code), using
- linear OA(390, 19682, F3, 15) (dual of [19682, 19592, 16]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using
- linear OA(39, 12, F3, 7) (dual of [12, 3, 8]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(399, 19694, F3, 15) (dual of [19694, 19595, 16]-code), using
(84, 84+15, 8083247)-Net in Base 3 — Upper bound on s
There is no (84, 99, 8083248)-net in base 3, because
- 1 times m-reduction [i] would yield (84, 98, 8083248)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 57264 200566 368126 152924 533115 403952 778484 654657 > 398 [i]