MinT - Best Known (31, 104, s)-Nets in Base 27

Best Known (31, 104, s)-Nets in Base 27

(31, 104, 114)-Net over F₂₇ — Constructive and digital

Digital (31, 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]

(31, 104, 160)-Net in Base 27 — Constructive

(31, 104, 160)-net in base 27, using

base change [i] based on digital (5, 78, 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]

(31, 104, 208)-Net over F₂₇ — Digital

Digital (31, 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]

(31, 104, 6821)-Net in Base 27 — Upper bound on s

There is no (31, 104, 6822)-net in base 27, because

1 times m-reduction [i] would yield (31, 103, 6822)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2707 201046 679941 451803 557007 503957 223128 701446 455717 277893 451490 235193 430162 904935 600339 303676 209753 513045 983932 202329 297308 468562 394514 782741 598537 > 27¹⁰³ [i]