MinT - Best Known (85, 85+17, s)-Nets in Base 32

Best Known (85, 85+17, s)-Nets in Base 32

(85, 85+17, 1049642)-Net over F₃₂ — Constructive and digital

Digital (85, 102, 1049642)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (13, 21, 1067)-net over F₃₂, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 33)-net over F₃₂, using
      - s-reduction based on digital (0, 0, s)-net over F₃₂ with arbitrarily large s, using
        arbitrary set of 32⁰ points [i]
    2. digital (0, 0, 33)-net over F₃₂ (see above)
    3. digital (0, 0, 33)-net over F₃₂ (see above)
    4. digital (0, 0, 33)-net over F₃₂ (see above)
    5. digital (0, 0, 33)-net over F₃₂ (see above)
    6. digital (0, 0, 33)-net over F₃₂ (see above)
    7. digital (0, 0, 33)-net over F₃₂ (see above)
    8. digital (0, 0, 33)-net over F₃₂ (see above)
    9. digital (0, 0, 33)-net over F₃₂ (see above)
    10. digital (0, 0, 33)-net over F₃₂ (see above)
    11. digital (0, 0, 33)-net over F₃₂ (see above)
    12. digital (0, 0, 33)-net over F₃₂ (see above)
    13. digital (0, 0, 33)-net over F₃₂ (see above)
    14. digital (0, 0, 33)-net over F₃₂ (see above)
    15. digital (0, 0, 33)-net over F₃₂ (see above)
    16. digital (0, 0, 33)-net over F₃₂ (see above)
    17. digital (0, 0, 33)-net over F₃₂ (see above)
    18. digital (0, 0, 33)-net over F₃₂ (see above)
    19. digital (0, 0, 33)-net over F₃₂ (see above)
    20. digital (0, 0, 33)-net over F₃₂ (see above)
    21. digital (0, 0, 33)-net over F₃₂ (see above)
    22. digital (0, 0, 33)-net over F₃₂ (see above)
    23. digital (0, 0, 33)-net over F₃₂ (see above)
    24. digital (0, 0, 33)-net over F₃₂ (see above)
    25. digital (0, 1, 33)-net over F₃₂, using
      - s-reduction based on digital (0, 1, s)-net over F₃₂ with arbitrarily large s, using
        construction for strength kÂ =Â 1 [i]
    26. digital (0, 1, 33)-net over F₃₂ (see above)
    27. digital (0, 1, 33)-net over F₃₂ (see above)
    28. digital (0, 1, 33)-net over F₃₂ (see above)
    29. digital (0, 2, 33)-net over F₃₂, using
      - kÂ =Â 2 construction [i]
    30. digital (0, 2, 33)-net over F₃₂ (see above)
    31. digital (0, 4, 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]
    32. digital (1, 9, 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]
2. digital (64, 81, 1048575)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁸¹, 1048575, F₃₂, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(32⁸¹, 8388601, F₃₂, 17) (dual of [8388601, 8388520, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁸¹, large, F₃₂, 17) (dual of [large, large−81, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(85, 85+17, 1064959)-Net in Base 32 — Constructive

(85, 102, 1064959)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (16, 24, 16384)-net in base 32, using
  - base change [i] based on digital (7, 15, 16384)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256¹⁵, 16384, F₂₅₆, 8, 8) (dual of [(16384, 8), 131057, 9]-NRT-code), using
      - OA 4-folding and stacking [i] based on linear OA(256¹⁵, 65536, F₂₅₆, 8) (dual of [65536, 65521, 9]-code), using
        an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
2. (61, 78, 1048575)-net in base 32, using
  - base change [i] based on digital (48, 65, 1048575)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁶⁵, 1048575, F₆₄, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
      - OOA 8-folding and stacking with additional row [i] based on linear OA(64⁶⁵, 8388601, F₆₄, 17) (dual of [8388601, 8388536, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁶⁵, large, F₆₄, 17) (dual of [large, large−65, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(85, 85+17, large)-Net over F₃₂ — Digital

Digital (85, 102, large)-net over F₃₂, using

t-expansion [i] based on digital (83, 102, large)-net over F₃₂, using
- 3 times m-reduction [i] based on digital (83, 105, large)-net over F₃₂, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(85, 85+17, large)-Net in Base 32 — Upper bound on s

There is no (85, 102, large)-net in base 32, because

15 times m-reduction [i] would yield (85, 87, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (85, 85+17, s)-Nets in Base 32

(85, 85+17, 1049642)-Net over F32 — Constructive and digital

(85, 85+17, 1064959)-Net in Base 32 — Constructive

(85, 85+17, large)-Net over F32 — Digital

(85, 85+17, large)-Net in Base 32 — Upper bound on s

(85, 85+17, 1049642)-Net over F₃₂ — Constructive and digital

(85, 85+17, large)-Net over F₃₂ — Digital