MinT - Best Known (71−25, 71, s)-Nets in Base 32

Best Known (71−25, 71, s)-Nets in Base 32

(71−25, 71, 341)-Net over F₃₂ — Constructive and digital

Digital (46, 71, 341)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 2, 33)-net over F₃₂, using
  - kÂ =Â 2 construction [i]
2. digital (0, 2, 33)-net over F₃₂ (see above)
3. digital (0, 3, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
      - the rational function field F₃₂(x) [i]
    - Niederreiter sequence [i]
4. digital (0, 3, 33)-net over F₃₂ (see above)
5. digital (0, 4, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
6. digital (0, 5, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
7. digital (0, 6, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
8. digital (0, 8, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
9. digital (0, 12, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
10. digital (1, 26, 44)-net over F₃₂, using
  - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
      - a function field by SÃ©mirat [i]

(71−25, 71, 1365)-Net in Base 32 — Constructive

(46, 71, 1365)-net in base 32, using

32¹ times duplication [i] based on (45, 70, 1365)-net in base 32, using
- base change [i] based on digital (25, 50, 1365)-net over F₁₂₈, using
  - 128¹ times duplication [i] based on digital (24, 49, 1365)-net over F₁₂₈, using
    - net defined by OOA [i] based on linear OOA(128⁴⁹, 1365, F₁₂₈, 25, 25) (dual of [(1365, 25), 34076, 26]-NRT-code), using
      - OOA 12-folding and stacking with additional row [i] based on linear OA(128⁴⁹, 16381, F₁₂₈, 25) (dual of [16381, 16332, 26]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁴⁹, 16384, F₁₂₈, 25) (dual of [16384, 16335, 26]-code), using
        an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(71−25, 71, 8981)-Net over F₃₂ — Digital

Digital (46, 71, 8981)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(71−25, 71, large)-Net in Base 32 — Upper bound on s

There is no (46, 71, large)-net in base 32, because

23 times m-reduction [i] would yield (46, 48, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (71−25, 71, s)-Nets in Base 32

(71−25, 71, 341)-Net over F32 — Constructive and digital

(71−25, 71, 1365)-Net in Base 32 — Constructive

(71−25, 71, 8981)-Net over F32 — Digital

(71−25, 71, large)-Net in Base 32 — Upper bound on s

(71−25, 71, 341)-Net over F₃₂ — Constructive and digital

(71−25, 71, 8981)-Net over F₃₂ — Digital