MinT - Best Known (69−35, 69, s)-Nets in Base 32

Best Known (69−35, 69, s)-Nets in Base 32

(69−35, 69, 202)-Net over F₃₂ — Constructive and digital

Digital (34, 69, 202)-net over F₃₂, using

1 times m-reduction [i] based on digital (34, 70, 202)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 25, 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, 45, 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]

(69−35, 69, 288)-Net in Base 32 — Constructive

(34, 69, 288)-net in base 32, using

t-expansion [i] based on (33, 69, 288)-net in base 32, using
- 15 times m-reduction [i] based on (33, 84, 288)-net in base 32, using
  - base change [i] based on digital (9, 60, 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]

(69−35, 69, 522)-Net over F₃₂ — Digital

Digital (34, 69, 522)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁶⁹, 522, F₃₂, 35) (dual of [522, 453, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁶⁹, 1023, F₃₂, 35) (dual of [1023, 954, 36]-code), using
  - the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {3,4,…,37}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [i]

(69−35, 69, 242754)-Net in Base 32 — Upper bound on s

There is no (34, 69, 242755)-net in base 32, because

1 times m-reduction [i] would yield (34, 68, 242755)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 239749 165192 492152 190404 542442 997219 771476 374054 167772 889157 493510 049583 728143 738864 892004 626632 802248 > 32⁶⁸ [i]