MinT - Best Known (9, 9+11, s)-Nets in Base 81

Best Known (9, 9+11, s)-Nets in Base 81

(9, 9+11, 264)-Net over F₈₁ — Constructive and digital

Digital (9, 20, 264)-net over F₈₁, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 3, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
      - the rational function field F₈₁(x) [i]
    - Niederreiter sequence [i]
2. digital (0, 5, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
3. digital (1, 12, 100)-net over F₈₁, using
  - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
      - a function field by SÃ©mirat [i]

(9, 9+11, 425)-Net over F₈₁ — Digital

Digital (9, 20, 425)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81²⁰, 425, F₈₁, 11) (dual of [425, 405, 12]-code), using
- 14 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 13 times 0) [i] based on linear OA(81¹⁹, 410, F₈₁, 11) (dual of [410, 391, 12]-code), using
  - the BCH-code C(I) with length 410Â | 81²âˆ’1, defining interval IÂ =Â {1,50,99,…,81}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(9, 9+11, 582086)-Net in Base 81 — Upper bound on s

There is no (9, 20, 582087)-net in base 81, because

1 times m-reduction [i] would yield (9, 19, 582087)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 1 824802 568552 487490 593136 634571 906801 > 81¹⁹ [i]