MinT - Best Known (146−30, 146, s)-Nets in Base 9

Best Known (146−30, 146, s)-Nets in Base 9

(146−30, 146, 3946)-Net over F₉ — Constructive and digital

Digital (116, 146, 3946)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 15, 10)-net over F₉, using
  - net from sequence [i] based on digital (0, 9)-sequence over F₉, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 0 and N(F) ≥ 10, using
      - the rational function field F₉(x) [i]
    - Niederreiter sequence [i]
2. digital (101, 131, 3936)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9¹³¹, 3936, F₉, 30, 30) (dual of [(3936, 30), 117949, 31]-NRT-code), using
    - OA 15-folding and stacking [i] based on linear OA(9¹³¹, 59040, F₉, 30) (dual of [59040, 58909, 31]-code), using
      - discarding factors / shortening the dual code based on linear OA(9¹³¹, 59049, F₉, 30) (dual of [59049, 58918, 31]-code), using
        an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 9⁵âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]

(146−30, 146, 92943)-Net over F₉ — Digital

Digital (116, 146, 92943)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(146−30, 146, large)-Net in Base 9 — Upper bound on s

There is no (116, 146, large)-net in base 9, because

28 times m-reduction [i] would yield (116, 118, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]