MinT - Best Known (103−55, 103, s)-Nets in Base 27

Best Known (103−55, 103, s)-Nets in Base 27

(103−55, 103, 190)-Net over F₂₇ — Constructive and digital

Digital (48, 103, 190)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 37, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]
2. digital (11, 66, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
      - an explicitly constructive algebraic function field [i]

(103−55, 103, 370)-Net in Base 27 — Constructive

(48, 103, 370)-net in base 27, using

t-expansion [i] based on (43, 103, 370)-net in base 27, using
- 5 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(103−55, 103, 428)-Net over F₂₇ — Digital

Digital (48, 103, 428)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(103−55, 103, 107337)-Net in Base 27 — Upper bound on s

There is no (48, 103, 107338)-net in base 27, because

1 times m-reduction [i] would yield (48, 102, 107338)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 99 810340 485341 062942 653070 309930 079013 899709 847353 621717 211173 659111 208977 675012 337096 786607 392666 344862 070516 651558 404990 726492 620566 665737 375025 > 27¹⁰² [i]