MinT - Best Known (30, 30+73, s)-Nets in Base 27

Best Known (30, 30+73, s)-Nets in Base 27

(30, 30+73, 114)-Net over F₂₇ — Constructive and digital

Digital (30, 103, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 103, 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]

(30, 30+73, 150)-Net in Base 27 — Constructive

(30, 103, 150)-net in base 27, using

1 times m-reduction [i] based on (30, 104, 150)-net in base 27, using
- base change [i] based on digital (4, 78, 150)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

(30, 30+73, 208)-Net over F₂₇ — Digital

Digital (30, 103, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 103, 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]

(30, 30+73, 6222)-Net in Base 27 — Upper bound on s

There is no (30, 103, 6223)-net in base 27, because

1 times m-reduction [i] would yield (30, 102, 6223)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 99 963200 321680 981892 458944 028239 824429 526576 953141 261497 101018 661696 304803 144935 169786 504635 710618 307999 562458 442623 904551 934122 262202 556021 651545 > 27¹⁰² [i]