Best Known (112−20, 112, s)-Nets in Base 3
(112−20, 112, 688)-Net over F3 — Constructive and digital
Digital (92, 112, 688)-net over F3, using
- t-expansion [i] based on digital (91, 112, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 28, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 28, 172)-net over F81, using
(112−20, 112, 3296)-Net over F3 — Digital
Digital (92, 112, 3296)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3112, 3296, F3, 2, 20) (dual of [(3296, 2), 6480, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3112, 6592, F3, 20) (dual of [6592, 6480, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3111, 6591, F3, 20) (dual of [6591, 6480, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(3105, 6561, F3, 20) (dual of [6561, 6456, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(36, 30, F3, 3) (dual of [30, 24, 4]-code or 30-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3111, 6591, F3, 20) (dual of [6591, 6480, 21]-code), using
- OOA 2-folding [i] based on linear OA(3112, 6592, F3, 20) (dual of [6592, 6480, 21]-code), using
(112−20, 112, 499684)-Net in Base 3 — Upper bound on s
There is no (92, 112, 499685)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 273894 061561 605509 733684 654215 432936 138377 385354 200241 > 3112 [i]