Best Known (20, 20+11, s)-Nets in Base 9
(20, 20+11, 300)-Net over F9 — Constructive and digital
Digital (20, 31, 300)-net over F9, using
- 1 times m-reduction [i] based on 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
- trace code for nets [i] based on digital (4, 16, 150)-net over F81, using
(20, 20+11, 750)-Net over F9 — Digital
Digital (20, 31, 750)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(931, 750, F9, 11) (dual of [750, 719, 12]-code), using
- 15 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 11 times 0) [i] based on linear OA(928, 732, F9, 11) (dual of [732, 704, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(928, 729, F9, 11) (dual of [729, 701, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(90, 3, F9, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- 15 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 11 times 0) [i] based on linear OA(928, 732, F9, 11) (dual of [732, 704, 12]-code), using
(20, 20+11, 173059)-Net in Base 9 — Upper bound on s
There is no (20, 31, 173060)-net in base 9, because
- 1 times m-reduction [i] would yield (20, 30, 173060)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 42392 118704 679693 340442 573217 > 930 [i]