MinT - Best Known (7, 7+9, s)-Nets in Base 27

Best Known (7, 7+9, s)-Nets in Base 27

(7, 7+9, 86)-Net over F₂₇ — Constructive and digital

Digital (7, 16, 86)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (1, 5, 38)-net over F₂₇, using
  - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. digital (2, 11, 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]

(7, 7+9, 116)-Net in Base 27 — Constructive

(7, 16, 116)-net in base 27, using

4 times m-reduction [i] based on (7, 20, 116)-net in base 27, using
- base change [i] based on digital (2, 15, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(7, 7+9, 149)-Net over F₂₇ — Digital

Digital (7, 16, 149)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27¹⁶, 149, F₂₇, 9) (dual of [149, 133, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(27¹⁶, 182, F₂₇, 9) (dual of [182, 166, 10]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 182Â | 27²âˆ’1, defining interval IÂ =Â [0,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(7, 7+9, 19845)-Net in Base 27 — Upper bound on s

There is no (7, 16, 19846)-net in base 27, because

1 times m-reduction [i] would yield (7, 15, 19846)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2954 737161 931512 655881 > 27¹⁵ [i]