Best Known (20, 20+12, s)-Nets in Base 9
(20, 20+12, 300)-Net over F9 — Constructive and digital
Digital (20, 32, 300)-net over F9, using
- trace code for nets [i] based on digital (4, 16, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
(20, 20+12, 509)-Net over F9 — Digital
Digital (20, 32, 509)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(932, 509, F9, 12) (dual of [509, 477, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(932, 736, F9, 12) (dual of [736, 704, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(931, 729, F9, 12) (dual of [729, 698, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(925, 729, F9, 10) (dual of [729, 704, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(91, 7, F9, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(932, 736, F9, 12) (dual of [736, 704, 13]-code), using
(20, 20+12, 45961)-Net in Base 9 — Upper bound on s
There is no (20, 32, 45962)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 3 433880 522451 869501 345158 592545 > 932 [i]