MinT - Best Known (90−61, 90, s)-Nets in Base 27

Best Known (90−61, 90, s)-Nets in Base 27

(90−61, 90, 114)-Net over F₂₇ — Constructive and digital

Digital (29, 90, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 90, 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]

(90−61, 90, 160)-Net in Base 27 — Constructive

(29, 90, 160)-net in base 27, using

6 times m-reduction [i] based on (29, 96, 160)-net in base 27, using
- base change [i] based on digital (5, 72, 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]

(90−61, 90, 208)-Net over F₂₇ — Digital

Digital (29, 90, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 90, 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]

(90−61, 90, 8153)-Net in Base 27 — Upper bound on s

There is no (29, 90, 8154)-net in base 27, because

1 times m-reduction [i] would yield (29, 89, 8154)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 24 639922 163409 181193 495082 139016 333039 827686 176940 072450 374079 864199 827685 471287 486205 858076 550102 115046 927676 032068 963213 508861 > 27⁸⁹ [i]