MinT - Best Known (29, 29+75, s)-Nets in Base 27

Best Known (29, 29+75, s)-Nets in Base 27

(29, 29+75, 114)-Net over F₂₇ — Constructive and digital

Digital (29, 104, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 104, 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]

(29, 29+75, 116)-Net in Base 27 — Constructive

(29, 104, 116)-net in base 27, using

4 times m-reduction [i] based on (29, 108, 116)-net in base 27, using
- base change [i] based on digital (2, 81, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(29, 29+75, 208)-Net over F₂₇ — Digital

Digital (29, 104, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 104, 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]

(29, 29+75, 5419)-Net in Base 27 — Upper bound on s

There is no (29, 104, 5420)-net in base 27, because

1 times m-reduction [i] would yield (29, 103, 5420)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2695 793926 946751 426835 606325 997377 858206 814727 400842 265231 504725 580190 081510 680241 739949 839872 262526 863808 514594 816332 470171 803089 816504 336698 356057 > 27¹⁰³ [i]