MinT - Best Known (105−73, 105, s)-Nets in Base 27

Best Known (105−73, 105, s)-Nets in Base 27

(105−73, 105, 114)-Net over F₂₇ — Constructive and digital

Digital (32, 105, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 105, 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]

(105−73, 105, 160)-Net in Base 27 — Constructive

(32, 105, 160)-net in base 27, using

3 times m-reduction [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]

(105−73, 105, 208)-Net over F₂₇ — Digital

Digital (32, 105, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 105, 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]

(105−73, 105, 7476)-Net in Base 27 — Upper bound on s

There is no (32, 105, 7477)-net in base 27, because

1 times m-reduction [i] would yield (32, 104, 7477)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 72795 999705 418344 709085 716987 669067 112400 579331 099761 282049 845270 301627 031210 279690 388570 241233 045700 493320 133250 756375 417202 361539 299217 666636 067057 > 27¹⁰⁴ [i]