MinT - Best Known (30, 105, s)-Nets in Base 27

Best Known (30, 105, s)-Nets in Base 27

(30, 105, 114)-Net over F₂₇ — Constructive and digital

Digital (30, 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]

(30, 105, 116)-Net in Base 27 — Constructive

(30, 105, 116)-net in base 27, using

t-expansion [i] based on (29, 105, 116)-net in base 27, using
- 3 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, 105, 208)-Net over F₂₇ — Digital

Digital (30, 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]

(30, 105, 5926)-Net in Base 27 — Upper bound on s

There is no (30, 105, 5927)-net in base 27, because

1 times m-reduction [i] would yield (30, 104, 5927)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 72896 021914 139979 833905 126023 773548 982316 318088 727484 604651 196584 025220 868114 565667 439956 149265 519492 490626 076928 892779 722128 149710 486216 605529 458063 > 27¹⁰⁴ [i]