Best Known (27, 27+15, s)-Nets in Base 9
(27, 27+15, 320)-Net over F9 — Constructive and digital
Digital (27, 42, 320)-net over F9, using
- 2 times m-reduction [i] based on digital (27, 44, 320)-net over F9, using
- trace code for nets [i] based on digital (5, 22, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 22, 160)-net over F81, using
(27, 27+15, 717)-Net over F9 — Digital
Digital (27, 42, 717)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(942, 717, F9, 15) (dual of [717, 675, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(942, 728, F9, 15) (dual of [728, 686, 16]-code), using
- the primitive narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(942, 728, F9, 15) (dual of [728, 686, 16]-code), using
(27, 27+15, 164041)-Net in Base 9 — Upper bound on s
There is no (27, 42, 164042)-net in base 9, because
- 1 times m-reduction [i] would yield (27, 41, 164042)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 1330 321384 785349 010814 945743 886233 918705 > 941 [i]