MinT - Best Known (91, 105, s)-Nets in Base 32

Best Known (91, 105, s)-Nets in Base 32

(91, 105, 2397798)-Net over F₃₂ — Constructive and digital

Digital (91, 105, 2397798)-net over F₃₂, using

generalized (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 (24, 31, 1198371)-net over F₃₂, using
  - s-reduction based on digital (24, 31, 2796200)-net over F₃₂, using
    - net defined by OOA [i] based on linear OOA(32³¹, 2796200, F₃₂, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OA(32³¹, 8388601, F₃₂, 7) (dual of [8388601, 8388570, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(32³¹, large, F₃₂, 7) (dual of [large, large−31, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
3. digital (52, 66, 1198371)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁶⁶, 1198371, F₃₂, 14, 14) (dual of [(1198371, 14), 16777128, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(32⁶⁶, 8388597, F₃₂, 14) (dual of [8388597, 8388531, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁶⁶, large, F₃₂, 14) (dual of [large, large−66, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(91, 105, 2429382)-Net in Base 32 — Constructive

(91, 105, 2429382)-net in base 32, using

32¹ times duplication [i] based on (90, 104, 2429382)-net in base 32, using
- base change [i] based on digital (51, 65, 2429382)-net over F₂₅₆, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (2, 6, 32640)-net over F₂₅₆, using
      - net defined by OOA [i] based on linear OOA(256⁶, 32640, F₂₅₆, 4, 4) (dual of [(32640, 4), 130554, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(256⁶, 65280, F₂₅₆, 4) (dual of [65280, 65274, 5]-code), using
        1 times truncation [i] based on linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 6]-code), using
        a code by Danev and Olsson [i]
    2. digital (12, 19, 1198371)-net over F₂₅₆, using
      - s-reduction based on digital (12, 19, 2796200)-net over F₂₅₆, using
        net defined by OOA [i] based on linear OOA(256¹⁹, 2796200, F₂₅₆, 7, 7) (dual of [(2796200, 7), 19573381, 8]-NRT-code), using
        appending kth column [i] based on linear OOA(256¹⁹, 2796200, F₂₅₆, 6, 7) (dual of [(2796200, 6), 16777181, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(256¹⁹, 8388601, F₂₅₆, 7) (dual of [8388601, 8388582, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(256¹⁹, large, F₂₅₆, 7) (dual of [large, large−19, 8]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
    3. digital (26, 40, 1198371)-net over F₂₅₆, using
      - net defined by OOA [i] based on linear OOA(256⁴⁰, 1198371, F₂₅₆, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
        OA 7-folding and stacking [i] based on linear OA(256⁴⁰, 8388597, F₂₅₆, 14) (dual of [8388597, 8388557, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁰, large, F₂₅₆, 14) (dual of [large, large−40, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(91, 105, large)-Net over F₃₂ — Digital

Digital (91, 105, large)-net over F₃₂, using

t-expansion [i] based on digital (87, 105, large)-net over F₃₂, using
- 5 times m-reduction [i] based on digital (87, 110, large)-net over F₃₂, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(91, 105, large)-Net in Base 32 — Upper bound on s

There is no (91, 105, large)-net in base 32, because

12 times m-reduction [i] would yield (91, 93, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (91, 105, s)-Nets in Base 32

(91, 105, 2397798)-Net over F32 — Constructive and digital

(91, 105, 2429382)-Net in Base 32 — Constructive

(91, 105, large)-Net over F32 — Digital

(91, 105, large)-Net in Base 32 — Upper bound on s

(91, 105, 2397798)-Net over F₃₂ — Constructive and digital

(91, 105, large)-Net over F₃₂ — Digital