MinT - Best Known (27, 27+63, s)-Nets in Base 32

Best Known (27, 27+63, s)-Nets in Base 32

(27, 27+63, 120)-Net over F₃₂ — Constructive and digital

Digital (27, 90, 120)-net over F₃₂, using

t-expansion [i] based on digital (11, 90, 120)-net over F₃₂, using
- net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
    - a function field by SÃ©mirat [i]

(27, 27+63, 177)-Net in Base 32 — Constructive

(27, 90, 177)-net in base 32, using

t-expansion [i] based on (25, 90, 177)-net in base 32, using
- 18 times m-reduction [i] based on (25, 108, 177)-net in base 32, using
  - base change [i] based on digital (7, 90, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(27, 27+63, 225)-Net over F₃₂ — Digital

Digital (27, 90, 225)-net over F₃₂, using

t-expansion [i] based on digital (24, 90, 225)-net over F₃₂, using
- net from sequence [i] based on digital (24, 224)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 24 and N(F) ≥ 225, using
    - a curve from the manYPoints database [i]

(27, 27+63, 257)-Net in Base 32

(27, 90, 257)-net in base 32, using

base change [i] based on digital (12, 75, 257)-net over F₆₄, using
- net from sequence [i] based on digital (12, 256)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 12 and N(F) ≥ 257, using
    - K_1,2 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

(27, 27+63, 8376)-Net in Base 32 — Upper bound on s

There is no (27, 90, 8377)-net in base 32, because

1 times m-reduction [i] would yield (27, 89, 8377)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 90 910502 020875 876197 692444 717739 026853 934808 819583 615745 598063 031468 877957 532611 679118 814731 025206 813888 295952 454832 638459 158997 781792 > 32⁸⁹ [i]