Best Known (56−21, 56, s)-Nets in Base 9
(56−21, 56, 344)-Net over F9 — Constructive and digital
Digital (35, 56, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 28, 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
(56−21, 56, 563)-Net over F9 — Digital
Digital (35, 56, 563)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(956, 563, F9, 21) (dual of [563, 507, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(956, 737, F9, 21) (dual of [737, 681, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(955, 730, F9, 21) (dual of [730, 675, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 96−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(949, 730, F9, 19) (dual of [730, 681, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 96−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [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 C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(956, 737, F9, 21) (dual of [737, 681, 22]-code), using
(56−21, 56, 100275)-Net in Base 9 — Upper bound on s
There is no (35, 56, 100276)-net in base 9, because
- 1 times m-reduction [i] would yield (35, 55, 100276)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 30433 778401 786586 568277 662561 874947 447793 523896 698561 > 955 [i]