MinT - Best Known (80−28, 80, s)-Nets in Base 32

Best Known (80−28, 80, s)-Nets in Base 32

(80−28, 80, 341)-Net over F₃₂ — Constructive and digital

Digital (52, 80, 341)-net over F₃₂, using

1 times m-reduction [i] based on digital (52, 81, 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, 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]
  3. digital (0, 3, 33)-net over F₃₂ (see above)
  4. digital (0, 4, 33)-net over F₃₂, using
    - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
  5. digital (0, 4, 33)-net 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, 7, 33)-net over F₃₂, using
    - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
  8. digital (0, 9, 33)-net over F₃₂, using
    - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
  9. digital (0, 14, 33)-net over F₃₂, using
    - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
  10. digital (1, 30, 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]

(80−28, 80, 1170)-Net in Base 32 — Constructive

(52, 80, 1170)-net in base 32, using

t-expansion [i] based on (51, 80, 1170)-net in base 32, using
- net defined by OOA [i] based on OOA(32⁸⁰, 1170, S₃₂, 29, 29), using
  - OOA 14-folding and stacking with additional row [i] based on OA(32⁸⁰, 16381, S₃₂, 29), using
    - discarding factors based on OA(32⁸⁰, 16386, S₃₂, 29), using
      - discarding parts of the base [i] based on linear OA(128⁵⁷, 16386, F₁₂₈, 29) (dual of [16386, 16329, 30]-code), using
        constructionÂ X applied to C_e(28) âŠ‚ C_e(27) [i] based on
        linear OA(128⁵⁷, 16384, F₁₂₈, 29) (dual of [16384, 16327, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
        linear OA(128⁵⁵, 16384, F₁₂₈, 28) (dual of [16384, 16329, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
        linear OA(128⁰, 2, F₁₂₈, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(80−28, 80, 10170)-Net over F₃₂ — Digital

Digital (52, 80, 10170)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(80−28, 80, large)-Net in Base 32 — Upper bound on s

There is no (52, 80, large)-net in base 32, because

26 times m-reduction [i] would yield (52, 54, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (80−28, 80, s)-Nets in Base 32

(80−28, 80, 341)-Net over F32 — Constructive and digital

(80−28, 80, 1170)-Net in Base 32 — Constructive

(80−28, 80, 10170)-Net over F32 — Digital

(80−28, 80, large)-Net in Base 32 — Upper bound on s

(80−28, 80, 341)-Net over F₃₂ — Constructive and digital

(80−28, 80, 10170)-Net over F₃₂ — Digital