MinT - Best Known (75−51, 75, s)-Nets in Base 27

Best Known (75−51, 75, s)-Nets in Base 27

(75−51, 75, 114)-Net over F₂₇ — Constructive and digital

Digital (24, 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]

(75−51, 75, 160)-Net in Base 27 — Constructive

(24, 75, 160)-net in base 27, using

1 times m-reduction [i] based on (24, 76, 160)-net in base 27, using
- base change [i] based on digital (5, 57, 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]

(75−51, 75, 208)-Net over F₂₇ — Digital

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]

(75−51, 75, 6739)-Net in Base 27 — Upper bound on s

There is no (24, 75, 6740)-net in base 27, because

1 times m-reduction [i] would yield (24, 74, 6740)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 8337 500746 814812 471554 727289 498503 460478 915363 212988 075273 432409 526005 626857 301284 003746 247460 161173 206921 > 27⁷⁴ [i]