MinT - Best Known (42−6, 42, s)-Nets in Base 27

Best Known (42−6, 42, s)-Nets in Base 27

(42−6, 42, large)-Net over F₂₇ — Constructive and digital

Digital (36, 42, large)-net over F₂₇, using

27¹ times duplication [i] based on digital (35, 41, 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, 0, 532899)-net over F₂₇ (see above)
  22. 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]
  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 (20, 26, 2796201)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27²⁶, 2796201, F₂₇, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(27²⁶, large, F₂₇, 6) (dual of [large, large−26, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(42−6, 42, large)-Net in Base 27 — Upper bound on s

There is no (36, 42, large)-net in base 27, because

4 times m-reduction [i] would yield (36, 38, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (42−6, 42, s)-Nets in Base 27

(42−6, 42, large)-Net over F27 — Constructive and digital

(42−6, 42, large)-Net in Base 27 — Upper bound on s

(42−6, 42, large)-Net over F₂₇ — Constructive and digital