Best Known (53−19, 53, s)-Nets in Base 9
(53−19, 53, 344)-Net over F9 — Constructive and digital
Digital (34, 53, 344)-net over F9, using
- 1 times m-reduction [i] based on digital (34, 54, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 27, 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, 27, 172)-net over F81, using
(53−19, 53, 735)-Net over F9 — Digital
Digital (34, 53, 735)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(953, 735, F9, 19) (dual of [735, 682, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(953, 742, F9, 19) (dual of [742, 689, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(949, 729, F9, 19) (dual of [729, 680, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(940, 729, F9, 15) (dual of [729, 689, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(94, 13, F9, 3) (dual of [13, 9, 4]-code or 13-cap in PG(3,9)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(953, 742, F9, 19) (dual of [742, 689, 20]-code), using
(53−19, 53, 169063)-Net in Base 9 — Upper bound on s
There is no (34, 53, 169064)-net in base 9, because
- 1 times m-reduction [i] would yield (34, 52, 169064)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 41 747030 966683 317299 030935 005464 789249 947439 673665 > 952 [i]