MinT - Best Known (29, 29+77, s)-Nets in Base 27

Best Known (29, 29+77, s)-Nets in Base 27

(29, 29+77, 114)-Net over F₂₇ — Constructive and digital

Digital (29, 106, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 106, 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]

(29, 29+77, 116)-Net in Base 27 — Constructive

(29, 106, 116)-net in base 27, using

2 times m-reduction [i] based on (29, 108, 116)-net in base 27, using
- base change [i] based on digital (2, 81, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(29, 29+77, 208)-Net over F₂₇ — Digital

Digital (29, 106, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 106, 208)-net over F₂₇, using
- net from sequence [i] based on digital (24, 207)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 24 and N(F) ≥ 208, using
    - an algebraic function field by Beelen and Pellikaan [i]

(29, 29+77, 5190)-Net in Base 27 — Upper bound on s

There is no (29, 106, 5191)-net in base 27, because

1 times m-reduction [i] would yield (29, 105, 5191)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 964681 524000 463366 239243 872303 418684 399199 012698 087108 024257 295825 539217 578349 091452 705317 225085 209653 318232 873608 777011 946586 138850 369117 458804 654453 > 27¹⁰⁵ [i]