MinT - Best Known (27, 64, s)-Nets in Base 32

Best Known (27, 64, s)-Nets in Base 32

(27, 64, 142)-Net over F₃₂ — Constructive and digital

Digital (27, 64, 142)-net over F₃₂, using

1 times m-reduction [i] based on digital (27, 65, 142)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 20, 44)-net over F₃₂, using
    - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
        a function field by SÃ©mirat [i]
  2. digital (7, 45, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(27, 64, 226)-Net over F₃₂ — Digital

Digital (27, 64, 226)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32⁶⁴, 226, F₃₂, 2, 37) (dual of [(226, 2), 388, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(32⁶⁴, 227, F₃₂, 2, 37) (dual of [(227, 2), 390, 38]-NRT-code), using
  - constructionÂ X applied to AG(2;F,410P) âŠ‚ AG(2;F,414P) [i] based on
    1. linear OOA(32⁶¹, 224, F₃₂, 2, 37) (dual of [(224, 2), 387, 38]-NRT-code), using algebraic-geometric NRT-code AG(2;F,410P) [i] based on function field F/F₃₂ with g(F) = 24 and N(F) ≥ 225, using
      - a curve from the manYPoints database [i]
    2. linear OOA(32⁵⁷, 224, F₃₂, 2, 33) (dual of [(224, 2), 391, 34]-NRT-code), using algebraic-geometric NRT-code AG(2;F,414P) [i] based on function field F/F₃₂ with g(F) = 24 and N(F) ≥ 225 (see above)
    3. linear OOA(32³, 3, F₃₂, 2, 3) (dual of [(3, 2), 3, 4]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(32³, 32, F₃₂, 2, 3) (dual of [(32, 2), 61, 4]-NRT-code), using
        Reedâ€“Solomon NRT-code RS(2;61,32) [i]

(27, 64, 260)-Net in Base 32 — Constructive

(27, 64, 260)-net in base 32, using

base change [i] based on digital (3, 40, 260)-net over F₂₅₆, using
- net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(27, 64, 321)-Net in Base 32

(27, 64, 321)-net in base 32, using

2 times m-reduction [i] based on (27, 66, 321)-net in base 32, using
- base change [i] based on (16, 55, 321)-net in base 64, using
  - 1 times m-reduction [i] based on (16, 56, 321)-net in base 64, using
    - base change [i] based on digital (2, 42, 321)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
        sharp bound on the number of rational points on curves with genus 2 [i]

(27, 64, 45155)-Net in Base 32 — Upper bound on s

There is no (27, 64, 45156)-net in base 32, because

1 times m-reduction [i] would yield (27, 63, 45156)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 66775 879716 329254 050197 267090 845396 785497 539791 778729 704009 048854 878468 100266 293739 257461 280950 > 32⁶³ [i]