MinT - Best Known (32, 109, s)-Nets in Base 27

Best Known (32, 109, s)-Nets in Base 27

(32, 109, 114)-Net over F₂₇ — Constructive and digital

Digital (32, 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]

(32, 109, 150)-Net in Base 27 — Constructive

(32, 109, 150)-net in base 27, using

27¹ times duplication [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]

(32, 109, 208)-Net over F₂₇ — Digital

Digital (32, 109, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 109, 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]

(32, 109, 6739)-Net in Base 27 — Upper bound on s

There is no (32, 109, 6740)-net in base 27, because

1 times m-reduction [i] would yield (32, 108, 6740)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 38753 904759 472336 134684 999178 526483 776525 373502 921076 260094 768787 036671 033451 005963 212463 027857 269462 825876 877655 046307 943578 855917 088881 877034 571270 111673 > 27¹⁰⁸ [i]