MinT - Best Known (109−79, 109, s)-Nets in Base 27

Best Known (109−79, 109, s)-Nets in Base 27

(109−79, 109, 114)-Net over F₂₇ — Constructive and digital

Digital (30, 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]

(109−79, 109, 116)-Net in Base 27 — Constructive

(30, 109, 116)-net in base 27, using

27¹ times duplication [i] based on (29, 108, 116)-net in base 27, using
- base change [i] based on digital (2, 81, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(109−79, 109, 208)-Net over F₂₇ — Digital

Digital (30, 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]

(109−79, 109, 5427)-Net in Base 27 — Upper bound on s

There is no (30, 109, 5428)-net in base 27, because

1 times m-reduction [i] would yield (30, 108, 5428)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 38826 099915 067751 791918 004415 724263 949542 100164 823023 423077 630638 917092 206578 869814 447492 087808 129096 190400 619610 360237 188673 867973 529473 398533 913362 127889 > 27¹⁰⁸ [i]