MinT - Best Known (82−14, 82, s)-Nets in Base 27

Best Known (82−14, 82, s)-Nets in Base 27

(82−14, 82, 1199127)-Net over F₂₇ — Constructive and digital

Digital (68, 82, 1199127)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (9, 16, 756)-net over F₂₇, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 28)-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, 28)-net over F₂₇ (see above)
    3. digital (0, 0, 28)-net over F₂₇ (see above)
    4. digital (0, 0, 28)-net over F₂₇ (see above)
    5. digital (0, 0, 28)-net over F₂₇ (see above)
    6. digital (0, 0, 28)-net over F₂₇ (see above)
    7. digital (0, 0, 28)-net over F₂₇ (see above)
    8. digital (0, 0, 28)-net over F₂₇ (see above)
    9. digital (0, 0, 28)-net over F₂₇ (see above)
    10. digital (0, 0, 28)-net over F₂₇ (see above)
    11. digital (0, 0, 28)-net over F₂₇ (see above)
    12. digital (0, 0, 28)-net over F₂₇ (see above)
    13. digital (0, 0, 28)-net over F₂₇ (see above)
    14. digital (0, 0, 28)-net over F₂₇ (see above)
    15. digital (0, 0, 28)-net over F₂₇ (see above)
    16. digital (0, 0, 28)-net over F₂₇ (see above)
    17. digital (0, 0, 28)-net over F₂₇ (see above)
    18. digital (0, 0, 28)-net over F₂₇ (see above)
    19. digital (0, 0, 28)-net over F₂₇ (see above)
    20. digital (0, 0, 28)-net over F₂₇ (see above)
    21. digital (0, 1, 28)-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, 28)-net over F₂₇ (see above)
    23. digital (0, 1, 28)-net over F₂₇ (see above)
    24. digital (0, 1, 28)-net over F₂₇ (see above)
    25. digital (0, 2, 28)-net over F₂₇, using
      - kÂ =Â 2 construction [i]
    26. digital (0, 3, 28)-net over F₂₇, using
      - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
        the rational function field F₂₇(x) [i]
        Niederreiter sequence [i]
    27. digital (0, 7, 28)-net over F₂₇, using
      - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
2. digital (52, 66, 1198371)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁶⁶, 1198371, F₂₇, 14, 14) (dual of [(1198371, 14), 16777128, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(27⁶⁶, 8388597, F₂₇, 14) (dual of [8388597, 8388531, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁶⁶, large, F₂₇, 14) (dual of [large, large−66, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(82−14, 82, large)-Net over F₂₇ — Digital

Digital (68, 82, large)-net over F₂₇, using

4 times m-reduction [i] based on digital (68, 86, large)-net over F₂₇, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁸⁶, large, F₂₇, 18) (dual of [large, large−86, 19]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(82−14, 82, large)-Net in Base 27 — Upper bound on s

There is no (68, 82, large)-net in base 27, because

12 times m-reduction [i] would yield (68, 70, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (82−14, 82, s)-Nets in Base 27

(82−14, 82, 1199127)-Net over F27 — Constructive and digital

(82−14, 82, large)-Net over F27 — Digital

(82−14, 82, large)-Net in Base 27 — Upper bound on s

(82−14, 82, 1199127)-Net over F₂₇ — Constructive and digital

(82−14, 82, large)-Net over F₂₇ — Digital