MinT - Best Known (109−72, 109, s)-Nets in Base 27

Best Known (109−72, 109, s)-Nets in Base 27

(109−72, 109, 114)-Net over F₂₇ — Constructive and digital

Digital (37, 109, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 109, 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]

(109−72, 109, 172)-Net in Base 27 — Constructive

(37, 109, 172)-net in base 27, using

27¹ times duplication [i] based on (36, 108, 172)-net in base 27, using
- t-expansion [i] based on (34, 108, 172)-net in base 27, using
  - base change [i] based on digital (7, 81, 172)-net over F₈₁, using
    - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
        a function field by SÃ©mirat [i]

(109−72, 109, 244)-Net over F₂₇ — Digital

Digital (37, 109, 244)-net over F₂₇, using

t-expansion [i] based on digital (36, 109, 244)-net over F₂₇, using
- net from sequence [i] based on digital (36, 243)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 36 and N(F) ≥ 244, using
    - Drinfeld modules of rank 1 [i]

(109−72, 109, 11828)-Net in Base 27 — Upper bound on s

There is no (37, 109, 11829)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 046082 518836 090798 796112 256703 548672 140585 295110 341248 353601 704741 258415 823083 006325 774013 661325 403723 748928 846179 512100 832856 500273 683588 466072 048927 811825 > 27¹⁰⁹ [i]