MinT - Best Known (86−38, 86, s)-Nets in Base 32

Best Known (86−38, 86, s)-Nets in Base 32

(86−38, 86, 260)-Net over F₃₂ — Constructive and digital

Digital (48, 86, 260)-net over F₃₂, using

generalized (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 (7, 26, 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, 45, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(86−38, 86, 513)-Net in Base 32 — Constructive

(48, 86, 513)-net in base 32, using

t-expansion [i] based on (46, 86, 513)-net in base 32, using
- 22 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]

(86−38, 86, 1508)-Net over F₃₂ — Digital

Digital (48, 86, 1508)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(86−38, 86, 1662016)-Net in Base 32 — Upper bound on s

There is no (48, 86, 1662017)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2772 675592 101880 694869 595070 151087 494990 394793 053114 702072 799106 319013 624818 431674 411985 070316 082309 569634 977902 851450 150470 197824 > 32⁸⁶ [i]