MinT - Best Known (45−9, 45, s)-Nets in Base 9

Best Known (45−9, 45, s)-Nets in Base 9

(45−9, 45, 14777)-Net over F₉ — Constructive and digital

Digital (36, 45, 14777)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 5, 16)-net over F₉, using
  - net from sequence [i] based on digital (1, 15)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 1 and N(F) ≥ 16, using
      - a function field by GarcÃa and Quoos [i]
2. digital (31, 40, 14761)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9⁴⁰, 14761, F₉, 9, 9) (dual of [(14761, 9), 132809, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(9⁴⁰, 59045, F₉, 9) (dual of [59045, 59005, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(9⁴⁰, 59048, F₉, 9) (dual of [59048, 59008, 10]-code), using
        1 times truncation [i] based on linear OA(9⁴¹, 59049, F₉, 10) (dual of [59049, 59008, 11]-code), using
        an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 9⁵âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(45−9, 45, 109706)-Net over F₉ — Digital

Digital (36, 45, 109706)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(45−9, 45, large)-Net in Base 9 — Upper bound on s

There is no (36, 45, large)-net in base 9, because

7 times m-reduction [i] would yield (36, 38, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]