MinT - Best Known (99−63, 99, s)-Nets in Base 27

Best Known (99−63, 99, s)-Nets in Base 27

(99−63, 99, 114)-Net over F₂₇ — Constructive and digital

Digital (36, 99, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 99, 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]

(99−63, 99, 172)-Net in Base 27 — Constructive

(36, 99, 172)-net in base 27, using

t-expansion [i] based on (34, 99, 172)-net in base 27, using
- 9 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]

(99−63, 99, 244)-Net over F₂₇ — Digital

Digital (36, 99, 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]

(99−63, 99, 15979)-Net in Base 27 — Upper bound on s

There is no (36, 99, 15980)-net in base 27, because

1 times m-reduction [i] would yield (36, 98, 15980)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 187 786363 731181 425083 838466 650165 361845 071640 572371 313813 709065 828868 744968 809371 399943 917712 510253 823935 289424 096706 629892 717573 890727 462577 > 27⁹⁸ [i]