MinT - Best Known (38, 99, s)-Nets in Base 27

Best Known (38, 99, s)-Nets in Base 27

(38, 99, 128)-Net over F₂₇ — Constructive and digital

Digital (38, 99, 128)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 34, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (4, 65, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(38, 99, 224)-Net in Base 27 — Constructive

(38, 99, 224)-net in base 27, using

1 times m-reduction [i] based on (38, 100, 224)-net in base 27, using
- base change [i] based on digital (13, 75, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(38, 99, 244)-Net over F₂₇ — Digital

Digital (38, 99, 244)-net over F₂₇, using

t-expansion [i] based on 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]

(38, 99, 298)-Net in Base 27

(38, 99, 298)-net in base 27, using

5 times m-reduction [i] based on (38, 104, 298)-net in base 27, using
- base change [i] based on digital (12, 78, 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]

(38, 99, 21942)-Net in Base 27 — Upper bound on s

There is no (38, 99, 21943)-net in base 27, because

1 times m-reduction [i] would yield (38, 98, 21943)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 187 806344 963904 561230 270036 265394 452566 912670 872796 904139 468673 116415 896506 135964 191101 573106 540370 669108 742766 409368 451299 925297 610060 123045 > 27⁹⁸ [i]