MinT - Best Known (107−69, 107, s)-Nets in Base 27

Best Known (107−69, 107, s)-Nets in Base 27

(107−69, 107, 114)-Net over F₂₇ — Constructive and digital

Digital (38, 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]

(107−69, 107, 172)-Net in Base 27 — Constructive

(38, 107, 172)-net in base 27, using

t-expansion [i] based on (34, 107, 172)-net in base 27, using
- 1 times m-reduction [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]

(107−69, 107, 244)-Net over F₂₇ — Digital

Digital (38, 107, 244)-net over F₂₇, using

t-expansion [i] based on digital (36, 107, 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]

(107−69, 107, 15082)-Net in Base 27 — Upper bound on s

There is no (38, 107, 15083)-net in base 27, because

1 times m-reduction [i] would yield (38, 106, 15083)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 53 079970 794196 803019 541708 574250 256387 424080 055399 813935 143106 121905 068078 617128 305582 138071 606183 269747 934312 541900 662811 669901 577378 586468 517607 823365 > 27¹⁰⁶ [i]