MinT - Best Known (9, 9+11, s)-Nets in Base 27

Best Known (9, 9+11, s)-Nets in Base 27

(9, 9+11, 96)-Net over F₂₇ — Constructive and digital

Digital (9, 20, 96)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (2, 7, 351)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁷, 351, F₂₇, 5, 5) (dual of [(351, 5), 1748, 6]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OA(27⁷, 703, F₂₇, 5) (dual of [703, 696, 6]-code), using
      - a code by Danev and Olsson [i]
2. digital (2, 13, 48)-net over F₂₇, using
  - net from sequence [i] based on digital (2, 47)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 2 and N(F) ≥ 48, using
      - a function field by GarcÃa and Quoos [i]

(9, 9+11, 150)-Net in Base 27 — Constructive

(9, 20, 150)-net in base 27, using

base change [i] based on digital (4, 15, 150)-net over F₈₁, using
- net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
    - a function field by SÃ©mirat [i]

(9, 9+11, 164)-Net over F₂₇ — Digital

Digital (9, 20, 164)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27²⁰, 164, F₂₇, 11) (dual of [164, 144, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(27²⁰, 182, F₂₇, 11) (dual of [182, 162, 12]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 182Â | 27²âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(9, 9+11, 27542)-Net in Base 27 — Upper bound on s

There is no (9, 20, 27543)-net in base 27, because

1 times m-reduction [i] would yield (9, 19, 27543)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1570 053797 331559 324956 400815 > 27¹⁹ [i]