MinT - Best Known (89−62, 89, s)-Nets in Base 27

Best Known (89−62, 89, s)-Nets in Base 27

(89−62, 89, 114)-Net over F₂₇ — Constructive and digital

Digital (27, 89, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 89, 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]

(89−62, 89, 150)-Net in Base 27 — Constructive

(27, 89, 150)-net in base 27, using

3 times m-reduction [i] based on (27, 92, 150)-net in base 27, using
- base change [i] based on digital (4, 69, 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]

(89−62, 89, 208)-Net over F₂₇ — Digital

Digital (27, 89, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 89, 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]

(89−62, 89, 6127)-Net in Base 27 — Upper bound on s

There is no (27, 89, 6128)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 24 634284 286382 142104 035837 050172 421281 941338 354539 497448 163778 963283 103454 610457 181559 627349 001979 009036 227977 608311 900676 632001 > 27⁸⁹ [i]