MinT - Best Known (86−4, 86, s)-Nets in Base 32

Best Known (86−4, 86, s)-Nets in Base 32

(86−4, 86, large)-Net over F₃₂ — Constructive and digital

Digital (82, 86, large)-net over F₃₂, using

t-expansion [i] based on digital (80, 86, large)-net over F₃₂, using
- 5 times m-reduction [i] based on digital (80, 91, large)-net over F₃₂, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 524289)-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, 524289)-net over F₃₂ (see above)
    3. digital (0, 0, 524289)-net over F₃₂ (see above)
    4. digital (0, 0, 524289)-net over F₃₂ (see above)
    5. digital (0, 0, 524289)-net over F₃₂ (see above)
    6. digital (0, 0, 524289)-net over F₃₂ (see above)
    7. digital (0, 0, 524289)-net over F₃₂ (see above)
    8. digital (0, 0, 524289)-net over F₃₂ (see above)
    9. digital (0, 0, 524289)-net over F₃₂ (see above)
    10. digital (0, 0, 524289)-net over F₃₂ (see above)
    11. digital (0, 0, 524289)-net over F₃₂ (see above)
    12. digital (0, 0, 524289)-net over F₃₂ (see above)
    13. digital (0, 0, 524289)-net over F₃₂ (see above)
    14. digital (0, 0, 524289)-net over F₃₂ (see above)
    15. digital (0, 0, 524289)-net over F₃₂ (see above)
    16. digital (0, 0, 524289)-net over F₃₂ (see above)
    17. digital (0, 0, 524289)-net over F₃₂ (see above)
    18. digital (0, 0, 524289)-net over F₃₂ (see above)
    19. digital (0, 0, 524289)-net over F₃₂ (see above)
    20. digital (0, 0, 524289)-net over F₃₂ (see above)
    21. digital (0, 0, 524289)-net over F₃₂ (see above)
    22. digital (0, 1, 524289)-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]
    23. digital (0, 1, 524289)-net over F₃₂ (see above)
    24. digital (0, 1, 524289)-net over F₃₂ (see above)
    25. digital (0, 1, 524289)-net over F₃₂ (see above)
    26. digital (0, 1, 524289)-net over F₃₂ (see above)
    27. digital (0, 1, 524289)-net over F₃₂ (see above)
    28. digital (3, 5, 524289)-net over F₃₂, using
      - s-reduction based on digital (3, 5, 1082401)-net over F₃₂, using
        kÂ =Â 2 construction [i]
    29. digital (3, 5, 524289)-net over F₃₂ (see above)
    30. digital (4, 7, 524289)-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 (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]
    32. 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]

(86−4, 86, large)-Net in Base 32 — Upper bound on s

There is no (82, 86, large)-net in base 32, because

2 times m-reduction [i] would yield (82, 84, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (86−4, 86, s)-Nets in Base 32

(86−4, 86, large)-Net over F32 — Constructive and digital

(86−4, 86, large)-Net in Base 32 — Upper bound on s

(86−4, 86, large)-Net over F₃₂ — Constructive and digital