Best Known (198, 198+36, s)-Nets in Base 3
(198, 198+36, 1496)-Net over F3 — Constructive and digital
Digital (198, 234, 1496)-net over F3, using
- 31 times duplication [i] based on digital (197, 233, 1496)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (7, 25, 16)-net over F3, using
- net from sequence [i] based on digital (7, 15)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 7 and N(F) ≥ 16, using
- net from sequence [i] based on digital (7, 15)-sequence over F3, using
- digital (172, 208, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 52, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 52, 370)-net over F81, using
- digital (7, 25, 16)-net over F3, using
- (u, u+v)-construction [i] based on
(198, 198+36, 12560)-Net over F3 — Digital
Digital (198, 234, 12560)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3234, 12560, F3, 36) (dual of [12560, 12326, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3234, 19754, F3, 36) (dual of [19754, 19520, 37]-code), using
- construction X applied to Ce(36) ⊂ Ce(27) [i] based on
- linear OA(3217, 19683, F3, 37) (dual of [19683, 19466, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(317, 71, F3, 7) (dual of [71, 54, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using
- construction X applied to Ce(36) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3234, 19754, F3, 36) (dual of [19754, 19520, 37]-code), using
(198, 198+36, 6021089)-Net in Base 3 — Upper bound on s
There is no (198, 234, 6021090)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 4429 694358 010127 971973 454479 041523 421810 473806 736112 254495 670915 488937 131401 145986 318736 050781 850553 292695 461621 > 3234 [i]