MinT - Best Known (75−50, 75, s)-Nets in Base 27

Best Known (75−50, 75, s)-Nets in Base 27

(75−50, 75, 114)-Net over F₂₇ — Constructive and digital

Digital (25, 75, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 75, 114)-net over F₂₇, using
- net from sequence [i] based on digital (23, 113)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 23 and N(F) ≥ 114, using
    - an explicitly constructive algebraic function field [i]

(75−50, 75, 160)-Net in Base 27 — Constructive

(25, 75, 160)-net in base 27, using

5 times m-reduction [i] based on (25, 80, 160)-net in base 27, using
- base change [i] based on digital (5, 60, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(75−50, 75, 208)-Net over F₂₇ — Digital

Digital (25, 75, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 75, 208)-net over F₂₇, using
- net from sequence [i] based on digital (24, 207)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 24 and N(F) ≥ 208, using
    - an algebraic function field by Beelen and Pellikaan [i]

(75−50, 75, 7691)-Net in Base 27 — Upper bound on s

There is no (25, 75, 7692)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 225394 313585 830924 922713 827866 241408 441666 309315 920416 533854 028783 743514 981024 356747 018644 169070 941897 949369 > 27⁷⁵ [i]