MinT - Best Known (78−27, 78, s)-Nets in Base 32

Best Known (78−27, 78, s)-Nets in Base 32

(78−27, 78, 374)-Net over F₃₂ — Constructive and digital

Digital (51, 78, 374)-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, 3, 33)-net over F₃₂ (see above)
6. digital (0, 4, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
7. digital (0, 5, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
8. digital (0, 6, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
9. digital (0, 9, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
10. digital (0, 13, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
11. digital (1, 28, 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]

(78−27, 78, 1260)-Net in Base 32 — Constructive

(51, 78, 1260)-net in base 32, using

32¹ times duplication [i] based on (50, 77, 1260)-net in base 32, using
- base change [i] based on digital (28, 55, 1260)-net over F₁₂₈, using
  - 128² times duplication [i] based on digital (26, 53, 1260)-net over F₁₂₈, using
    - net defined by OOA [i] based on linear OOA(128⁵³, 1260, F₁₂₈, 27, 27) (dual of [(1260, 27), 33967, 28]-NRT-code), using
      - OOA 13-folding and stacking with additional row [i] based on linear OA(128⁵³, 16381, F₁₂₈, 27) (dual of [16381, 16328, 28]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁵³, 16384, F₁₂₈, 27) (dual of [16384, 16331, 28]-code), using
        an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]

(78−27, 78, 11166)-Net over F₃₂ — Digital

Digital (51, 78, 11166)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(78−27, 78, large)-Net in Base 32 — Upper bound on s

There is no (51, 78, large)-net in base 32, because

25 times m-reduction [i] would yield (51, 53, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (78−27, 78, s)-Nets in Base 32

(78−27, 78, 374)-Net over F32 — Constructive and digital

(78−27, 78, 1260)-Net in Base 32 — Constructive

(78−27, 78, 11166)-Net over F32 — Digital

(78−27, 78, large)-Net in Base 32 — Upper bound on s

(78−27, 78, 374)-Net over F₃₂ — Constructive and digital

(78−27, 78, 11166)-Net over F₃₂ — Digital