MinT - Best Known (110−59, 110, s)-Nets in Base 27

Best Known (110−59, 110, s)-Nets in Base 27

(110−59, 110, 192)-Net over F₂₇ — Constructive and digital

Digital (51, 110, 192)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (11, 40, 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]
2. digital (11, 70, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(110−59, 110, 370)-Net in Base 27 — Constructive

(51, 110, 370)-net in base 27, using

27² times duplication [i] based on (49, 108, 370)-net in base 27, using
- t-expansion [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]

(110−59, 110, 440)-Net over F₂₇ — Digital

Digital (51, 110, 440)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(110−59, 110, 107656)-Net in Base 27 — Upper bound on s

There is no (51, 110, 107657)-net in base 27, because

1 times m-reduction [i] would yield (51, 109, 107657)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 043956 338304 604462 879669 493903 393785 358908 147253 793175 502921 924604 166207 304955 833673 051037 046614 149976 150783 880070 057476 005793 463667 327524 852667 151150 746139 > 27¹⁰⁹ [i]