Best Known (58−20, 58, s)-Nets in Base 9
(58−20, 58, 344)-Net over F9 — Constructive and digital
Digital (38, 58, 344)-net over F9, using
- 4 times m-reduction [i] based on digital (38, 62, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 31, 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, 31, 172)-net over F81, using
(58−20, 58, 836)-Net over F9 — Digital
Digital (38, 58, 836)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(958, 836, F9, 20) (dual of [836, 778, 21]-code), using
- 98 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 9 times 0, 1, 26 times 0, 1, 56 times 0) [i] based on linear OA(952, 732, F9, 20) (dual of [732, 680, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(952, 729, F9, 20) (dual of [729, 677, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- 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(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(19) ⊂ Ce(18) [i] based on
- 98 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 9 times 0, 1, 26 times 0, 1, 56 times 0) [i] based on linear OA(952, 732, F9, 20) (dual of [732, 680, 21]-code), using
(58−20, 58, 193856)-Net in Base 9 — Upper bound on s
There is no (38, 58, 193857)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 22 186077 525857 839033 470431 152729 564411 280607 832990 898065 > 958 [i]