MinT - Best Known (69−7, 69, s)-Nets in Base 32

Best Known (69−7, 69, s)-Nets in Base 32

(69−7, 69, large)-Net over F₃₂ — Constructive and digital

Digital (62, 69, large)-net over F₃₂, using

32⁶ times duplication [i] based on digital (56, 63, large)-net over F₃₂, using
- t-expansion [i] based on digital (54, 63, large)-net over F₃₂, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 262144)-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, 262144)-net over F₃₂ (see above)
    3. digital (0, 0, 262144)-net over F₃₂ (see above)
    4. digital (0, 0, 262144)-net over F₃₂ (see above)
    5. digital (0, 0, 262144)-net over F₃₂ (see above)
    6. digital (0, 0, 262144)-net over F₃₂ (see above)
    7. digital (0, 0, 262144)-net over F₃₂ (see above)
    8. digital (0, 0, 262144)-net over F₃₂ (see above)
    9. digital (0, 0, 262144)-net over F₃₂ (see above)
    10. digital (0, 0, 262144)-net over F₃₂ (see above)
    11. digital (0, 0, 262144)-net over F₃₂ (see above)
    12. digital (0, 0, 262144)-net over F₃₂ (see above)
    13. digital (0, 0, 262144)-net over F₃₂ (see above)
    14. digital (0, 0, 262144)-net over F₃₂ (see above)
    15. digital (0, 0, 262144)-net over F₃₂ (see above)
    16. digital (0, 0, 262144)-net over F₃₂ (see above)
    17. digital (0, 0, 262144)-net over F₃₂ (see above)
    18. digital (0, 0, 262144)-net over F₃₂ (see above)
    19. digital (0, 0, 262144)-net over F₃₂ (see above)
    20. digital (0, 0, 262144)-net over F₃₂ (see above)
    21. digital (0, 0, 262144)-net over F₃₂ (see above)
    22. digital (0, 0, 262144)-net over F₃₂ (see above)
    23. digital (0, 0, 262144)-net over F₃₂ (see above)
    24. digital (0, 1, 262144)-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]
    25. digital (0, 1, 262144)-net over F₃₂ (see above)
    26. digital (0, 1, 262144)-net over F₃₂ (see above)
    27. digital (0, 1, 262144)-net over F₃₂ (see above)
    28. digital (0, 1, 262144)-net over F₃₂ (see above)
    29. digital (3, 5, 262144)-net over F₃₂, using
      - s-reduction based on digital (3, 5, 1082401)-net over F₃₂, using
        kÂ =Â 2 construction [i]
    30. digital (4, 7, 262144)-net over F₃₂, using
      - s-reduction based on digital (4, 7, 1050624)-net over F₃₂, using
        net defined by OOA [i] based on linear OOA(32⁷, 1050624, F₃₂, 3, 3) (dual of [(1050624, 3), 3151865, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(32⁷, 1050624, F₃₂, 2, 3) (dual of [(1050624, 2), 2101241, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(32⁷, 1050624, F₃₂, 3) (dual of [1050624, 1050617, 4]-code or 1050624-cap in PG(6,32)), using
        product of two caps with tangent hyperplane [i]
    31. digital (9, 13, 262144)-net over F₃₂, using
      - s-reduction based on digital (9, 13, 524290)-net over F₃₂, using
        net defined by OOA [i] based on linear OOA(32¹³, 524290, F₃₂, 4, 4) (dual of [(524290, 4), 2097147, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(32¹³, 1048580, F₃₂, 4) (dual of [1048580, 1048567, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        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(32⁹, 1048576, F₃₂, 3) (dual of [1048576, 1048567, 4]-code or 1048576-cap in PG(8,32)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [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]
    32. digital (24, 33, 262144)-net over F₃₂, using
      - net defined by OOA [i] based on linear OOA(32³³, 262144, F₃₂, 9, 9) (dual of [(262144, 9), 2359263, 10]-NRT-code), using
        OOA 4-folding and stacking with additional row [i] based on linear OA(32³³, 1048577, F₃₂, 9) (dual of [1048577, 1048544, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 32⁸âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(69−7, 69, large)-Net in Base 32 — Upper bound on s

There is no (62, 69, large)-net in base 32, because

5 times m-reduction [i] would yield (62, 64, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (69−7, 69, s)-Nets in Base 32

(69−7, 69, large)-Net over F32 — Constructive and digital

(69−7, 69, large)-Net in Base 32 — Upper bound on s

(69−7, 69, large)-Net over F₃₂ — Constructive and digital