MinT - Best Known (99−69, 99, s)-Nets in Base 27

Best Known (99−69, 99, s)-Nets in Base 27

(99−69, 99, 114)-Net over F₂₇ — Constructive and digital

Digital (30, 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]

(99−69, 99, 160)-Net in Base 27 — Constructive

(30, 99, 160)-net in base 27, using

1 times m-reduction [i] based on (30, 100, 160)-net in base 27, using
- base change [i] based on digital (5, 75, 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]

(99−69, 99, 208)-Net over F₂₇ — Digital

Digital (30, 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]

(99−69, 99, 6935)-Net in Base 27 — Upper bound on s

There is no (30, 99, 6936)-net in base 27, because

1 times m-reduction [i] would yield (30, 98, 6936)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 188 138937 669318 344818 671028 101563 944700 669861 124514 611278 702329 646055 553612 960014 219199 244530 176168 490507 593556 626782 019914 183048 410456 409361 > 27⁹⁸ [i]