MinT - Best Known (18, 43, s)-Nets in Base 32

Best Known (18, 43, s)-Nets in Base 32

(18, 43, 128)-Net over F₃₂ — Constructive and digital

Digital (18, 43, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 15, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (3, 28, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(18, 43, 162)-Net over F₃₂ — Digital

Digital (18, 43, 162)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(18, 43, 258)-Net in Base 32 — Constructive

(18, 43, 258)-net in base 32, using

32¹ times duplication [i] based on (17, 42, 258)-net in base 32, using
- base change [i] based on (10, 35, 258)-net in base 64, using
  - 1 times m-reduction [i] based on (10, 36, 258)-net in base 64, using
    - base change [i] based on digital (1, 27, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(18, 43, 289)-Net in Base 32

(18, 43, 289)-net in base 32, using

32¹ times duplication [i] based on (17, 42, 289)-net in base 32, using
- base change [i] based on (10, 35, 289)-net in base 64, using
  - 1 times m-reduction [i] based on (10, 36, 289)-net in base 64, using
    - base change [i] based on digital (1, 27, 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]

(18, 43, 31618)-Net in Base 32 — Upper bound on s

There is no (18, 43, 31619)-net in base 32, because

1 times m-reduction [i] would yield (18, 42, 31619)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1645 701724 021333 659303 989905 900842 203035 208818 465789 922489 788576 > 32⁴² [i]