Best Known (73−20, 73, s)-Nets in Base 9
(73−20, 73, 740)-Net over F9 — Constructive and digital
Digital (53, 73, 740)-net over F9, using
- 1 times m-reduction [i] based on digital (53, 74, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 37, 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, 37, 370)-net over F81, using
(73−20, 73, 6184)-Net over F9 — Digital
Digital (53, 73, 6184)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(973, 6184, F9, 20) (dual of [6184, 6111, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(973, 6577, F9, 20) (dual of [6577, 6504, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(969, 6561, F9, 20) (dual of [6561, 6492, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(957, 6561, F9, 16) (dual of [6561, 6504, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(94, 16, F9, 3) (dual of [16, 12, 4]-code or 16-cap in PG(3,9)), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(973, 6577, F9, 20) (dual of [6577, 6504, 21]-code), using
(73−20, 73, 5234270)-Net in Base 9 — Upper bound on s
There is no (53, 73, 5234271)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 4567 763912 760872 536523 181880 979698 591253 323841 153475 152421 851466 375601 > 973 [i]