MinT - Best Known (99−67, 99, s)-Nets in Base 27

Best Known (99−67, 99, s)-Nets in Base 27

(99−67, 99, 114)-Net over F₂₇ — Constructive and digital

Digital (32, 99, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 99, 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]

(99−67, 99, 172)-Net in Base 27 — Constructive

(32, 99, 172)-net in base 27, using

1 times m-reduction [i] based on (32, 100, 172)-net in base 27, using
- base change [i] based on digital (7, 75, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(99−67, 99, 208)-Net over F₂₇ — Digital

Digital (32, 99, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 99, 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]

(99−67, 99, 9000)-Net in Base 27 — Upper bound on s

There is no (32, 99, 9001)-net in base 27, because

1 times m-reduction [i] would yield (32, 98, 9001)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 188 143224 244201 060330 558153 478486 186813 858003 154112 634061 631327 937736 062256 387746 465251 182989 175781 701536 348085 707735 540701 131922 564984 890027 > 27⁹⁸ [i]