MinT - Best Known (42−24, 42, s)-Nets in Base 32

Best Known (42−24, 42, s)-Nets in Base 32

(42−24, 42, 128)-Net over F₃₂ — Constructive and digital

Digital (18, 42, 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, 27, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(42−24, 42, 178)-Net over F₃₂ — Digital

Digital (18, 42, 178)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(42−24, 42, 259)-Net in Base 32 — Constructive

(18, 42, 259)-net in base 32, using

base change [i] based on (11, 35, 259)-net in base 64, using
- 1 times m-reduction [i] based on (11, 36, 259)-net in base 64, using
  - base change [i] based on digital (2, 27, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(42−24, 42, 321)-Net in Base 32

(18, 42, 321)-net in base 32, using

base change [i] based on (11, 35, 321)-net in base 64, using
- 1 times m-reduction [i] based on (11, 36, 321)-net in base 64, using
  - base change [i] based on digital (2, 27, 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]

(42−24, 42, 31618)-Net in Base 32 — Upper bound on s

There is no (18, 42, 31619)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1645 701724 021333 659303 989905 900842 203035 208818 465789 922489 788576 > 32⁴² [i]