MinT - Best Known (81−16, 81, s)-Nets in Base 9

Best Known (81−16, 81, s)-Nets in Base 9

(81−16, 81, 7401)-Net over F₉ — Constructive and digital

Digital (65, 81, 7401)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 10, 20)-net over F₉, using
  - net from sequence [i] based on digital (2, 19)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 2 and N(F) ≥ 20, using
      - a function field by GarcÃa and Quoos [i]
2. digital (55, 71, 7381)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9⁷¹, 7381, F₉, 16, 16) (dual of [(7381, 16), 118025, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(9⁷¹, 59048, F₉, 16) (dual of [59048, 58977, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(9⁷¹, 59049, F₉, 16) (dual of [59049, 58978, 17]-code), using
        an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 9⁵âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(81−16, 81, 114186)-Net over F₉ — Digital

Digital (65, 81, 114186)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(81−16, 81, large)-Net in Base 9 — Upper bound on s

There is no (65, 81, large)-net in base 9, because

14 times m-reduction [i] would yield (65, 67, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]