MinT - Best Known (35, 35+75, s)-Nets in Base 27

Best Known (35, 35+75, s)-Nets in Base 27

(35, 35+75, 114)-Net over F₂₇ — Constructive and digital

Digital (35, 110, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 110, 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]

(35, 35+75, 160)-Net in Base 27 — Constructive

(35, 110, 160)-net in base 27, using

27² times duplication [i] based on (33, 108, 160)-net in base 27, using
- t-expansion [i] based on (32, 108, 160)-net in base 27, using
  - base change [i] based on digital (5, 81, 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]

(35, 35+75, 220)-Net over F₂₇ — Digital

Digital (35, 110, 220)-net over F₂₇, using

t-expansion [i] based on digital (33, 110, 220)-net over F₂₇, using
- net from sequence [i] based on digital (33, 219)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 33 and N(F) ≥ 220, using
    - Drinfeld modules of rank 1 [i]

(35, 35+75, 9262)-Net in Base 27 — Upper bound on s

There is no (35, 110, 9263)-net in base 27, because

1 times m-reduction [i] would yield (35, 109, 9263)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 044075 002580 691847 596619 063801 968131 889698 801239 030489 509697 904450 904363 620912 158717 395016 675372 495791 087076 006098 401078 867746 116246 034959 121872 066773 644703 > 27¹⁰⁹ [i]