Best Known (24−9, 24, s)-Nets in Base 3
(24−9, 24, 56)-Net over F3 — Constructive and digital
Digital (15, 24, 56)-net over F3, using
- trace code for nets [i] based on digital (3, 12, 28)-net over F9, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- the Hermitian function field over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
(24−9, 24, 58)-Net over F3 — Digital
Digital (15, 24, 58)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(324, 58, F3, 9) (dual of [58, 34, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(324, 80, F3, 9) (dual of [80, 56, 10]-code), using
- the primitive narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(324, 80, F3, 9) (dual of [80, 56, 10]-code), using
(24−9, 24, 609)-Net in Base 3 — Upper bound on s
There is no (15, 24, 610)-net in base 3, because
- 1 times m-reduction [i] would yield (15, 23, 610)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 94437 264681 > 323 [i]