MinT - Best Known (69−45, 69, s)-Nets in Base 27

Best Known (69−45, 69, s)-Nets in Base 27

(69−45, 69, 114)-Net over F₂₇ — Constructive and digital

Digital (24, 69, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 69, 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]

(69−45, 69, 160)-Net in Base 27 — Constructive

(24, 69, 160)-net in base 27, using

7 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]

(69−45, 69, 208)-Net over F₂₇ — Digital

Digital (24, 69, 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]

(69−45, 69, 9237)-Net in Base 27 — Upper bound on s

There is no (24, 69, 9238)-net in base 27, because

1 times m-reduction [i] would yield (24, 68, 9238)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 21 532106 218248 637353 857675 668975 186955 875611 254641 524756 716270 671849 209181 923899 835367 907297 805717 > 27⁶⁸ [i]