MinT - Best Known (30, 30+77, s)-Nets in Base 27

Best Known (30, 30+77, s)-Nets in Base 27

(30, 30+77, 114)-Net over F₂₇ — Constructive and digital

Digital (30, 107, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 107, 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]

(30, 30+77, 116)-Net in Base 27 — Constructive

(30, 107, 116)-net in base 27, using

t-expansion [i] based on (29, 107, 116)-net in base 27, using
- 1 times m-reduction [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]

(30, 30+77, 208)-Net over F₂₇ — Digital

Digital (30, 107, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 107, 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]

(30, 30+77, 5663)-Net in Base 27 — Upper bound on s

There is no (30, 107, 5664)-net in base 27, because

1 times m-reduction [i] would yield (30, 106, 5664)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 53 353718 108823 108996 187071 230162 378214 063760 732273 591311 887877 153736 912671 469845 888530 211139 189808 024283 674845 848886 688352 764168 187585 305954 220837 157057 > 27¹⁰⁶ [i]