MinT - Best Known (35, 71, s)-Nets in Base 32

Best Known (35, 71, s)-Nets in Base 32

(35, 71, 202)-Net over F₃₂ — Constructive and digital

Digital (35, 71, 202)-net over F₃₂, using

2 times m-reduction [i] based on digital (35, 73, 202)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. 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]
  2. digital (9, 47, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(35, 71, 288)-Net in Base 32 — Constructive

(35, 71, 288)-net in base 32, using

20 times m-reduction [i] based on (35, 91, 288)-net in base 32, using
- base change [i] based on digital (9, 65, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

(35, 71, 533)-Net over F₃₂ — Digital

Digital (35, 71, 533)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁷¹, 533, F₃₂, 36) (dual of [533, 462, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁷¹, 1023, F₃₂, 36) (dual of [1023, 952, 37]-code), using
  - the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {3,4,…,38}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]

(35, 71, 210731)-Net in Base 32 — Upper bound on s

There is no (35, 71, 210732)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 73393 080114 525649 512307 897688 587224 028481 072381 230826 819170 704787 187749 828995 714129 109857 964819 120404 539230 > 32⁷¹ [i]