MinT - Best Known (64−35, 64, s)-Nets in Base 27

Best Known (64−35, 64, s)-Nets in Base 27

(64−35, 64, 152)-Net over F₂₇ — Constructive and digital

Digital (29, 64, 152)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 23, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
2. digital (6, 41, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(64−35, 64, 224)-Net in Base 27 — Constructive

(29, 64, 224)-net in base 27, using

base change [i] based on digital (13, 48, 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]

(64−35, 64, 260)-Net over F₂₇ — Digital

Digital (29, 64, 260)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(64−35, 64, 298)-Net in Base 27

(29, 64, 298)-net in base 27, using

4 times m-reduction [i] based on (29, 68, 298)-net in base 27, using
- base change [i] based on digital (12, 51, 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]

(64−35, 64, 55637)-Net in Base 27 — Upper bound on s

There is no (29, 64, 55638)-net in base 27, because

1 times m-reduction [i] would yield (29, 63, 55638)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 499485 544481 082628 390987 977838 885690 185177 342060 401416 730180 489078 893274 894264 937435 720349 > 27⁶³ [i]