MinT - Best Known (31, 31+67, s)-Nets in Base 27

Best Known (31, 31+67, s)-Nets in Base 27

(31, 31+67, 114)-Net over F₂₇ — Constructive and digital

Digital (31, 98, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 98, 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]

(31, 31+67, 160)-Net in Base 27 — Constructive

(31, 98, 160)-net in base 27, using

6 times m-reduction [i] based on (31, 104, 160)-net in base 27, using
- base change [i] based on digital (5, 78, 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]

(31, 31+67, 208)-Net over F₂₇ — Digital

Digital (31, 98, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 98, 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]

(31, 31+67, 8143)-Net in Base 27 — Upper bound on s

There is no (31, 98, 8144)-net in base 27, because

1 times m-reduction [i] would yield (31, 97, 8144)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 6 973082 025078 142641 332952 503193 702329 534323 106130 431085 731562 825218 459259 390926 930591 237309 986179 096927 452031 879652 853339 406219 969051 752225 > 27⁹⁷ [i]