MinT - Best Known (78−53, 78, s)-Nets in Base 32

Best Known (78−53, 78, s)-Nets in Base 32

(78−53, 78, 120)-Net over F₃₂ — Constructive and digital

Digital (25, 78, 120)-net over F₃₂, using

t-expansion [i] based on digital (11, 78, 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]

(78−53, 78, 177)-Net in Base 32 — Constructive

(25, 78, 177)-net in base 32, using

30 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]

(78−53, 78, 225)-Net over F₃₂ — Digital

Digital (25, 78, 225)-net over F₃₂, using

t-expansion [i] based on digital (24, 78, 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]

(78−53, 78, 257)-Net in Base 32

(25, 78, 257)-net in base 32, using

base change [i] based on digital (12, 65, 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]

(78−53, 78, 9747)-Net in Base 32 — Upper bound on s

There is no (25, 78, 9748)-net in base 32, because

1 times m-reduction [i] would yield (25, 77, 9748)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 78 890434 458059 350630 670369 118917 941573 636486 129145 095291 064398 283728 130987 457948 298439 038737 229993 058233 202130 735424 > 32⁷⁷ [i]