MinT - Best Known (67−11, 67, s)-Nets in Base 32

Best Known (67−11, 67, s)-Nets in Base 32

(67−11, 67, 1711512)-Net over F₃₂ — Constructive and digital

Digital (56, 67, 1711512)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (11, 16, 33792)-net over F₃₂, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 1056)-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, 1056)-net over F₃₂ (see above)
    3. digital (0, 0, 1056)-net over F₃₂ (see above)
    4. digital (0, 0, 1056)-net over F₃₂ (see above)
    5. digital (0, 0, 1056)-net over F₃₂ (see above)
    6. digital (0, 0, 1056)-net over F₃₂ (see above)
    7. digital (0, 0, 1056)-net over F₃₂ (see above)
    8. digital (0, 0, 1056)-net over F₃₂ (see above)
    9. digital (0, 0, 1056)-net over F₃₂ (see above)
    10. digital (0, 0, 1056)-net over F₃₂ (see above)
    11. digital (0, 0, 1056)-net over F₃₂ (see above)
    12. digital (0, 0, 1056)-net over F₃₂ (see above)
    13. digital (0, 0, 1056)-net over F₃₂ (see above)
    14. digital (0, 0, 1056)-net over F₃₂ (see above)
    15. digital (0, 0, 1056)-net over F₃₂ (see above)
    16. digital (0, 0, 1056)-net over F₃₂ (see above)
    17. digital (0, 0, 1056)-net over F₃₂ (see above)
    18. digital (0, 0, 1056)-net over F₃₂ (see above)
    19. digital (0, 0, 1056)-net over F₃₂ (see above)
    20. digital (0, 0, 1056)-net over F₃₂ (see above)
    21. digital (0, 0, 1056)-net over F₃₂ (see above)
    22. digital (0, 0, 1056)-net over F₃₂ (see above)
    23. digital (0, 0, 1056)-net over F₃₂ (see above)
    24. digital (0, 0, 1056)-net over F₃₂ (see above)
    25. digital (0, 0, 1056)-net over F₃₂ (see above)
    26. digital (0, 0, 1056)-net over F₃₂ (see above)
    27. digital (0, 0, 1056)-net over F₃₂ (see above)
    28. digital (0, 1, 1056)-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]
    29. digital (0, 1, 1056)-net over F₃₂ (see above)
    30. digital (0, 1, 1056)-net over F₃₂ (see above)
    31. digital (1, 3, 1056)-net over F₃₂, using
      - s-reduction based on digital (1, 3, 1057)-net over F₃₂, using
        kÂ =Â 2 construction [i]
    32. digital (5, 10, 1056)-net over F₃₂, using
      - generalized (u,Â u+v)-construction [i] based on
        digital (0, 0, 33)-net over F₃₂, using
        s-reduction based on digital (0, 0, s)-net over F₃₂ with arbitrarily large s (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 0, 33)-net over F₃₂ (see above)
        digital (0, 1, 33)-net over F₃₂, using
        s-reduction based on digital (0, 1, s)-net over F₃₂ with arbitrarily large s (see above)
        digital (0, 1, 33)-net over F₃₂ (see above)
        digital (0, 1, 33)-net over F₃₂ (see above)
        digital (0, 2, 33)-net over F₃₂, using
        kÂ =Â 2 construction [i]
        digital (0, 5, 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 (40, 51, 1677720)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁵¹, 1677720, F₃₂, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(32⁵¹, 8388601, F₃₂, 11) (dual of [8388601, 8388550, 12]-code), using
      - discarding factors / shortening the dual code 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]

(67−11, 67, 2202009)-Net in Base 32 — Constructive

(56, 67, 2202009)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (12, 17, 524289)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32¹⁷, 524289, F₃₂, 5, 5) (dual of [(524289, 5), 2621428, 6]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OA(32¹⁷, 1048579, F₃₂, 5) (dual of [1048579, 1048562, 6]-code), using
      - discarding factors / shortening the dual code based on linear OA(32¹⁷, 1048580, F₃₂, 5) (dual of [1048580, 1048563, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(32¹⁷, 1048576, F₃₂, 5) (dual of [1048576, 1048559, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(32¹³, 1048576, F₃₂, 4) (dual of [1048576, 1048563, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(32⁰, 4, F₃₂, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(32⁰, s, F₃₂, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. (39, 50, 1677720)-net in base 32, using
  - net defined by OOA [i] based on OOA(32⁵⁰, 1677720, S₃₂, 11, 11), using
    - OOA 5-folding and stacking with additional row [i] based on OA(32⁵⁰, 8388601, S₃₂, 11), using
      - discarding factors based on OA(32⁵⁰, large, S₃₂, 11), using
        discarding parts of the base [i] based on linear OA(64⁴¹, large, F₆₄, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(67−11, 67, large)-Net over F₃₂ — Digital

Digital (56, 67, large)-net over F₃₂, using

4 times m-reduction [i] based on digital (56, 71, large)-net over F₃₂, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁷¹, large, F₃₂, 15) (dual of [large, large−71, 16]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(67−11, 67, large)-Net in Base 32 — Upper bound on s

There is no (56, 67, large)-net in base 32, because

9 times m-reduction [i] would yield (56, 58, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (67−11, 67, s)-Nets in Base 32

(67−11, 67, 1711512)-Net over F32 — Constructive and digital

(67−11, 67, 2202009)-Net in Base 32 — Constructive

(67−11, 67, large)-Net over F32 — Digital

(67−11, 67, large)-Net in Base 32 — Upper bound on s

(67−11, 67, 1711512)-Net over F₃₂ — Constructive and digital

(67−11, 67, large)-Net over F₃₂ — Digital