MinT - Best Known (77−33, 77, s)-Nets in Base 32

Best Known (77−33, 77, s)-Nets in Base 32

(77−33, 77, 260)-Net over F₃₂ — Constructive and digital

Digital (44, 77, 260)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 14, 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 (7, 23, 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]
3. digital (7, 40, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(77−33, 77, 513)-Net in Base 32 — Constructive

(44, 77, 513)-net in base 32, using

19 times m-reduction [i] based on (44, 96, 513)-net in base 32, using
- base change [i] based on digital (28, 80, 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]

(77−33, 77, 1743)-Net over F₃₂ — Digital

Digital (44, 77, 1743)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(77−33, 77, 3094840)-Net in Base 32 — Upper bound on s

There is no (44, 77, 3094841)-net in base 32, because

1 times m-reduction [i] would yield (44, 76, 3094841)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 462628 906221 334086 142256 642740 616050 457750 899588 112860 409567 594420 759614 072148 446451 375661 130206 180983 745039 962521 > 32⁷⁶ [i]