MinT - Best Known (40, 49, s)-Nets in Base 32

Best Known (40, 49, s)-Nets in Base 32

(40, 49, 2098206)-Net over F₃₂ — Constructive and digital

Digital (40, 49, 2098206)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (4, 8, 1056)-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, 0, 33)-net over F₃₂ (see above)
    26. digital (0, 0, 33)-net over F₃₂ (see above)
    27. digital (0, 0, 33)-net over F₃₂ (see above)
    28. digital (0, 0, 33)-net over F₃₂ (see above)
    29. 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]
    30. digital (0, 1, 33)-net over F₃₂ (see above)
    31. digital (0, 2, 33)-net over F₃₂, using
      - kÂ =Â 2 construction [i]
    32. 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]
2. digital (32, 41, 2097150)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁴¹, 2097150, F₃₂, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(32⁴¹, 8388601, F₃₂, 9) (dual of [8388601, 8388560, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁴¹, large, F₃₂, 9) (dual of [large, large−41, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(40, 49, 2105278)-Net in Base 32 — Constructive

(40, 49, 2105278)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (5, 9, 8128)-net in base 32, using
  - net defined by OOA [i] based on OOA(32⁹, 8128, S₃₂, 4, 4), using
    - OA 2-folding and stacking [i] based on OA(32⁹, 16256, S₃₂, 4), using
      - discarding parts of the base [i] based on linear OA(128⁶, 16256, F₁₂₈, 4) (dual of [16256, 16250, 5]-code), using
        1 times truncation [i] based on linear OA(128⁷, 16257, F₁₂₈, 5) (dual of [16257, 16250, 6]-code), using
        a code by Danev and Olsson [i]
2. (31, 40, 2097150)-net in base 32, using
  - base change [i] based on digital (16, 25, 2097150)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256²⁵, 2097150, F₂₅₆, 9, 9) (dual of [(2097150, 9), 18874325, 10]-NRT-code), using
      - OOA 4-folding and stacking with additional row [i] based on linear OA(256²⁵, 8388601, F₂₅₆, 9) (dual of [8388601, 8388576, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(256²⁵, large, F₂₅₆, 9) (dual of [large, large−25, 10]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(40, 49, large)-Net over F₃₂ — Digital

Digital (40, 49, large)-net over F₃₂, using

2 times m-reduction [i] based on digital (40, 51, large)-net over F₃₂, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁵¹, large, F₃₂, 11) (dual of [large, large−51, 12]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(40, 49, large)-Net in Base 32 — Upper bound on s

There is no (40, 49, large)-net in base 32, because

7 times m-reduction [i] would yield (40, 42, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (40, 49, s)-Nets in Base 32

(40, 49, 2098206)-Net over F32 — Constructive and digital

(40, 49, 2105278)-Net in Base 32 — Constructive

(40, 49, large)-Net over F32 — Digital

(40, 49, large)-Net in Base 32 — Upper bound on s

(40, 49, 2098206)-Net over F₃₂ — Constructive and digital

(40, 49, large)-Net over F₃₂ — Digital