MinT - Best Known (26, 26+65, s)-Nets in Base 27

Best Known (26, 26+65, s)-Nets in Base 27

(26, 26+65, 114)-Net over F₂₇ — Constructive and digital

Digital (26, 91, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 91, 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]

(26, 26+65, 116)-Net in Base 27 — Constructive

(26, 91, 116)-net in base 27, using

5 times m-reduction [i] based on (26, 96, 116)-net in base 27, using
- base change [i] based on digital (2, 72, 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]

(26, 26+65, 208)-Net over F₂₇ — Digital

Digital (26, 91, 208)-net over F₂₇, using

t-expansion [i] based on digital (24, 91, 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]

(26, 26+65, 5202)-Net in Base 27 — Upper bound on s

There is no (26, 91, 5203)-net in base 27, because

1 times m-reduction [i] would yield (26, 90, 5203)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 666 320008 779158 239710 842539 712280 462026 671534 591441 553011 369387 456570 493395 064235 423653 573084 036681 346709 996027 271809 896770 043585 > 27⁹⁰ [i]