MinT - Best Known (51, 61, s)-Nets in Base 27

Best Known (51, 61, s)-Nets in Base 27

(51, 61, 1687953)-Net over F₂₇ — Constructive and digital

Digital (51, 61, 1687953)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 15, 10233)-net over F₂₇, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 379)-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, 379)-net over F₂₇ (see above)
    3. digital (0, 0, 379)-net over F₂₇ (see above)
    4. digital (0, 0, 379)-net over F₂₇ (see above)
    5. digital (0, 0, 379)-net over F₂₇ (see above)
    6. digital (0, 0, 379)-net over F₂₇ (see above)
    7. digital (0, 0, 379)-net over F₂₇ (see above)
    8. digital (0, 0, 379)-net over F₂₇ (see above)
    9. digital (0, 0, 379)-net over F₂₇ (see above)
    10. digital (0, 0, 379)-net over F₂₇ (see above)
    11. digital (0, 0, 379)-net over F₂₇ (see above)
    12. digital (0, 0, 379)-net over F₂₇ (see above)
    13. digital (0, 0, 379)-net over F₂₇ (see above)
    14. digital (0, 0, 379)-net over F₂₇ (see above)
    15. digital (0, 0, 379)-net over F₂₇ (see above)
    16. digital (0, 0, 379)-net over F₂₇ (see above)
    17. digital (0, 0, 379)-net over F₂₇ (see above)
    18. digital (0, 0, 379)-net over F₂₇ (see above)
    19. digital (0, 0, 379)-net over F₂₇ (see above)
    20. digital (0, 0, 379)-net over F₂₇ (see above)
    21. digital (0, 0, 379)-net over F₂₇ (see above)
    22. digital (0, 0, 379)-net over F₂₇ (see above)
    23. digital (0, 1, 379)-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]
    24. digital (0, 1, 379)-net over F₂₇ (see above)
    25. digital (0, 1, 379)-net over F₂₇ (see above)
    26. digital (1, 3, 379)-net over F₂₇, using
      - s-reduction based on digital (1, 3, 757)-net over F₂₇, using
        kÂ =Â 2 construction [i]
    27. digital (4, 9, 379)-net over F₂₇, using
      - (u,Â u+v)-construction [i] based on
        digital (0, 2, 28)-net over F₂₇, using
        kÂ =Â 2 construction [i]
        digital (2, 7, 351)-net over F₂₇, using
        net defined by OOA [i] based on linear OOA(27⁷, 351, F₂₇, 5, 5) (dual of [(351, 5), 1748, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(27⁷, 703, F₂₇, 5) (dual of [703, 696, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (36, 46, 1677720)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁴⁶, 1677720, F₂₇, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(27⁴⁶, 8388600, F₂₇, 10) (dual of [8388600, 8388554, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁴⁶, large, F₂₇, 10) (dual of [large, large−46, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(51, 61, large)-Net over F₂₇ — Digital

Digital (51, 61, large)-net over F₂₇, using

t-expansion [i] based on digital (48, 61, large)-net over F₂₇, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁶¹, large, F₂₇, 13) (dual of [large, large−61, 14]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(51, 61, large)-Net in Base 27 — Upper bound on s

There is no (51, 61, large)-net in base 27, because

8 times m-reduction [i] would yield (51, 53, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (51, 61, s)-Nets in Base 27

(51, 61, 1687953)-Net over F27 — Constructive and digital

(51, 61, large)-Net over F27 — Digital

(51, 61, large)-Net in Base 27 — Upper bound on s

(51, 61, 1687953)-Net over F₂₇ — Constructive and digital

(51, 61, large)-Net over F₂₇ — Digital