Best Known (70, 70+27, s)-Nets in Base 9
(70, 70+27, 740)-Net over F9 — Constructive and digital
Digital (70, 97, 740)-net over F9, using
- 11 times m-reduction [i] based on digital (70, 108, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 54, 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, 54, 370)-net over F81, using
(70, 70+27, 5858)-Net over F9 — Digital
Digital (70, 97, 5858)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(997, 5858, F9, 27) (dual of [5858, 5761, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(997, 6562, F9, 27) (dual of [6562, 6465, 28]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 98−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(997, 6562, F9, 27) (dual of [6562, 6465, 28]-code), using
(70, 70+27, 7889140)-Net in Base 9 — Upper bound on s
There is no (70, 97, 7889141)-net in base 9, because
- 1 times m-reduction [i] would yield (70, 96, 7889141)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 40 483803 386729 209125 970799 220643 212245 323595 179813 728855 865322 595871 815986 817391 445547 288585 > 996 [i]