MinT - Best Known (27−15, 27, s)-Nets in Base 32

Best Known (27−15, 27, s)-Nets in Base 32

(27−15, 27, 120)-Net over F₃₂ — Constructive and digital

Digital (12, 27, 120)-net over F₃₂, using

t-expansion [i] based on digital (11, 27, 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]

(27−15, 27, 163)-Net over F₃₂ — Digital

Digital (12, 27, 163)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(27−15, 27, 258)-Net in Base 32 — Constructive

(12, 27, 258)-net in base 32, using

1 times m-reduction [i] based on (12, 28, 258)-net in base 32, using
- base change [i] based on (4, 20, 258)-net in base 128, using
  - 4 times m-reduction [i] based on (4, 24, 258)-net in base 128, using
    - base change [i] based on digital (1, 21, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(27−15, 27, 289)-Net in Base 32

(12, 27, 289)-net in base 32, using

1 times m-reduction [i] based on (12, 28, 289)-net in base 32, using
- base change [i] based on (4, 20, 289)-net in base 128, using
  - 4 times m-reduction [i] based on (4, 24, 289)-net in base 128, using
    - base change [i] based on digital (1, 21, 289)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
        sharp bound on the number of rational points on curves with genus 1 [i]

(27−15, 27, 42469)-Net in Base 32 — Upper bound on s

There is no (12, 27, 42470)-net in base 32, because

1 times m-reduction [i] would yield (12, 26, 42470)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1361 156438 225960 602262 503240 999495 028400 > 32²⁶ [i]