Best Known (81, 109, s)-Nets in Base 3
(81, 109, 252)-Net over F3 — Constructive and digital
Digital (81, 109, 252)-net over F3, using
- 31 times duplication [i] based on digital (80, 108, 252)-net over F3, using
- trace code for nets [i] based on digital (8, 36, 84)-net over F27, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 8 and N(F) ≥ 84, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- trace code for nets [i] based on digital (8, 36, 84)-net over F27, using
(81, 109, 484)-Net over F3 — Digital
Digital (81, 109, 484)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3109, 484, F3, 28) (dual of [484, 375, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3109, 728, F3, 28) (dual of [728, 619, 29]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,27], and designed minimum distance d ≥ |I|+1 = 29 [i]
- discarding factors / shortening the dual code based on linear OA(3109, 728, F3, 28) (dual of [728, 619, 29]-code), using
(81, 109, 15659)-Net in Base 3 — Upper bound on s
There is no (81, 109, 15660)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 10146 903568 399541 514582 653642 608969 383671 888199 230281 > 3109 [i]