MinT - Best Known (100−63, 100, s)-Nets in Base 27

Best Known (100−63, 100, s)-Nets in Base 27

(100−63, 100, 114)-Net over F₂₇ — Constructive and digital

Digital (37, 100, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 100, 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]

(100−63, 100, 172)-Net in Base 27 — Constructive

(37, 100, 172)-net in base 27, using

t-expansion [i] based on (34, 100, 172)-net in base 27, using
- 8 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]

(100−63, 100, 244)-Net over F₂₇ — Digital

Digital (37, 100, 244)-net over F₂₇, using

t-expansion [i] based on digital (36, 100, 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]

(100−63, 100, 298)-Net in Base 27

(37, 100, 298)-net in base 27, using

base change [i] based on digital (12, 75, 298)-net over F₈₁, using
- net from sequence [i] based on digital (12, 297)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 12 and N(F) ≥ 298, using
    - a curve from the manYPoints database [i]

(100−63, 100, 17774)-Net in Base 27 — Upper bound on s

There is no (37, 100, 17775)-net in base 27, because

1 times m-reduction [i] would yield (37, 99, 17775)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 5075 819891 017677 518275 475794 442127 129302 953582 197006 923457 876553 103851 544096 523127 608364 396822 110440 511823 981832 090754 240872 061850 514247 073307 > 27⁹⁹ [i]