MinT - Best Known (44−3, 44, s)-Nets in Base 27

Best Known (44−3, 44, s)-Nets in Base 27

(44−3, 44, large)-Net over F₂₇ — Constructive and digital

Digital (41, 44, large)-net over F₂₇, using

t-expansion [i] based on digital (40, 44, large)-net over F₂₇, using
- 3 times m-reduction [i] based on digital (40, 47, large)-net over F₂₇, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 532899)-net over F₂₇, using
      - s-reduction based on digital (0, 0, s)-net over F₂₇ with arbitrarily large s, using
        arbitrary set of 27⁰ points [i]
    2. digital (0, 0, 532899)-net over F₂₇ (see above)
    3. digital (0, 0, 532899)-net over F₂₇ (see above)
    4. digital (0, 0, 532899)-net over F₂₇ (see above)
    5. digital (0, 0, 532899)-net over F₂₇ (see above)
    6. digital (0, 0, 532899)-net over F₂₇ (see above)
    7. digital (0, 0, 532899)-net over F₂₇ (see above)
    8. digital (0, 0, 532899)-net over F₂₇ (see above)
    9. digital (0, 0, 532899)-net over F₂₇ (see above)
    10. digital (0, 0, 532899)-net over F₂₇ (see above)
    11. digital (0, 0, 532899)-net over F₂₇ (see above)
    12. digital (0, 0, 532899)-net over F₂₇ (see above)
    13. digital (0, 0, 532899)-net over F₂₇ (see above)
    14. digital (0, 0, 532899)-net over F₂₇ (see above)
    15. digital (0, 0, 532899)-net over F₂₇ (see above)
    16. digital (0, 0, 532899)-net over F₂₇ (see above)
    17. digital (0, 0, 532899)-net over F₂₇ (see above)
    18. digital (0, 0, 532899)-net over F₂₇ (see above)
    19. digital (0, 0, 532899)-net over F₂₇ (see above)
    20. digital (0, 0, 532899)-net over F₂₇ (see above)
    21. digital (0, 1, 532899)-net over F₂₇, using
      - s-reduction based on digital (0, 1, s)-net over F₂₇ with arbitrarily large s, using
        construction for strength kÂ =Â 1 [i]
    22. digital (0, 1, 532899)-net over F₂₇ (see above)
    23. digital (0, 1, 532899)-net over F₂₇ (see above)
    24. digital (0, 1, 532899)-net over F₂₇ (see above)
    25. digital (3, 5, 532899)-net over F₂₇, using
      - s-reduction based on digital (3, 5, 551881)-net over F₂₇, using
        kÂ =Â 2 construction [i]
    26. digital (4, 7, 532899)-net over F₂₇, using
      - net defined by OOA [i] based on linear OOA(27⁷, 532899, F₂₇, 3, 3) (dual of [(532899, 3), 1598690, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(27⁷, 532899, F₂₇, 2, 3) (dual of [(532899, 2), 1065791, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(27⁷, 532899, F₂₇, 3) (dual of [532899, 532892, 4]-code or 532899-cap in PG(6,27)), using
        product of two caps with tangent hyperplane [i]
    27. digital (24, 31, 2796200)-net over F₂₇, using
      - net defined by OOA [i] based on linear OOA(27³¹, 2796200, F₂₇, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(27³¹, 8388601, F₂₇, 7) (dual of [8388601, 8388570, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(27³¹, large, F₂₇, 7) (dual of [large, large−31, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(44−3, 44, large)-Net in Base 27 — Upper bound on s

There is no (41, 44, large)-net in base 27, because

1 times m-reduction [i] would yield (41, 43, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (44−3, 44, s)-Nets in Base 27

(44−3, 44, large)-Net over F27 — Constructive and digital

(44−3, 44, large)-Net in Base 27 — Upper bound on s

(44−3, 44, large)-Net over F₂₇ — Constructive and digital