MinT - Best Known (31, 31+75, s)-Nets in Base 27

Best Known (31, 31+75, s)-Nets in Base 27

(31, 31+75, 114)-Net over F₂₇ — Constructive and digital

Digital (31, 106, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 106, 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]

(31, 31+75, 150)-Net in Base 27 — Constructive

(31, 106, 150)-net in base 27, using

2 times m-reduction [i] based on (31, 108, 150)-net in base 27, using
- base change [i] based on digital (4, 81, 150)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

(31, 31+75, 208)-Net over F₂₇ — Digital

Digital (31, 106, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 106, 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]

(31, 31+75, 6480)-Net in Base 27 — Upper bound on s

There is no (31, 106, 6481)-net in base 27, because

1 times m-reduction [i] would yield (31, 105, 6481)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 968272 822035 364238 705621 692899 681318 541905 097858 060624 715453 010433 954179 388679 739757 101761 654198 475510 560797 628660 266970 156708 805468 295504 259645 688587 > 27¹⁰⁵ [i]