Best Known (101−38, 101, s)-Nets in Base 9
(101−38, 101, 344)-Net over F9 — Constructive and digital
Digital (63, 101, 344)-net over F9, using
- 11 times m-reduction [i] based on digital (63, 112, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 56, 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, 56, 172)-net over F81, using
(101−38, 101, 759)-Net over F9 — Digital
Digital (63, 101, 759)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9101, 759, F9, 38) (dual of [759, 658, 39]-code), using
- 26 step Varšamov–Edel lengthening with (ri) = (1, 25 times 0) [i] based on linear OA(9100, 732, F9, 38) (dual of [732, 632, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- linear OA(9100, 729, F9, 38) (dual of [729, 629, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(997, 729, F9, 37) (dual of [729, 632, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(37) ⊂ Ce(36) [i] based on
- 26 step Varšamov–Edel lengthening with (ri) = (1, 25 times 0) [i] based on linear OA(9100, 732, F9, 38) (dual of [732, 632, 39]-code), using
(101−38, 101, 117121)-Net in Base 9 — Upper bound on s
There is no (63, 101, 117122)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 2 390823 906211 937878 281406 395250 839854 034548 953455 464996 037472 970316 455596 060795 702405 372032 712049 > 9101 [i]