MinT - Best Known (105−44, 105, s)-Nets in Base 32

Best Known (105−44, 105, s)-Nets in Base 32

(105−44, 105, 316)-Net over F₃₂ — Constructive and digital

Digital (61, 105, 316)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 21, 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]
2. digital (7, 29, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
3. digital (11, 55, 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]

(105−44, 105, 513)-Net in Base 32 — Constructive

(61, 105, 513)-net in base 32, using

t-expansion [i] based on (46, 105, 513)-net in base 32, using
- 3 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
  - base change [i] based on digital (28, 90, 513)-net over F₆₄, using
    - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(105−44, 105, 2601)-Net over F₃₂ — Digital

Digital (61, 105, 2601)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(105−44, 105, 4458210)-Net in Base 32 — Upper bound on s

There is no (61, 105, 4458211)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 109 836912 293369 303312 079094 543450 258754 562013 018270 898579 295587 687983 854144 781922 797424 990633 713233 755670 641744 340180 791536 617622 764420 956553 731255 388086 504936 > 32¹⁰⁵ [i]