MinT - Best Known (109−73, 109, s)-Nets in Base 27

Best Known (109−73, 109, s)-Nets in Base 27

(109−73, 109, 114)-Net over F₂₇ — Constructive and digital

Digital (36, 109, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 109, 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]

(109−73, 109, 172)-Net in Base 27 — Constructive

(36, 109, 172)-net in base 27, using

27¹ times duplication [i] based on (35, 108, 172)-net in base 27, using
- t-expansion [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]

(109−73, 109, 244)-Net over F₂₇ — Digital

Digital (36, 109, 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]

(109−73, 109, 10791)-Net in Base 27 — Upper bound on s

There is no (36, 109, 10792)-net in base 27, because

1 times m-reduction [i] would yield (36, 108, 10792)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 38681 130175 760283 871481 536255 637311 930184 444435 307047 262832 560672 871429 571152 212284 663935 749428 190285 908947 768575 323524 428170 479436 020523 240925 631180 750401 > 27¹⁰⁸ [i]