MinT - Best Known (80, 96, s)-Nets in Base 27

Best Known (80, 96, s)-Nets in Base 27

(80, 96, 1049331)-Net over F₂₇ — Constructive and digital

Digital (80, 96, 1049331)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (12, 20, 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, 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]
    21. digital (0, 1, 28)-net over F₂₇ (see above)
    22. digital (0, 1, 28)-net over F₂₇ (see above)
    23. digital (0, 1, 28)-net over F₂₇ (see above)
    24. digital (0, 2, 28)-net over F₂₇, using
      - kÂ =Â 2 construction [i]
    25. digital (0, 2, 28)-net over F₂₇ (see above)
    26. digital (0, 4, 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, 8, 28)-net over F₂₇, using
      - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
2. digital (60, 76, 1048575)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁷⁶, 1048575, F₂₇, 16, 16) (dual of [(1048575, 16), 16777124, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(27⁷⁶, 8388600, F₂₇, 16) (dual of [8388600, 8388524, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁷⁶, large, F₂₇, 16) (dual of [large, large−76, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(80, 96, 1050215)-Net in Base 27 — Constructive

(80, 96, 1050215)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. (12, 20, 1640)-net in base 27, using
  - base change [i] based on digital (7, 15, 1640)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81¹⁵, 1640, F₈₁, 8, 8) (dual of [(1640, 8), 13105, 9]-NRT-code), using
      - OA 4-folding and stacking [i] based on linear OA(81¹⁵, 6560, F₈₁, 8) (dual of [6560, 6545, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(81¹⁵, 6561, F₈₁, 8) (dual of [6561, 6546, 9]-code), using
        an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
2. digital (60, 76, 1048575)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁷⁶, 1048575, F₂₇, 16, 16) (dual of [(1048575, 16), 16777124, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(27⁷⁶, 8388600, F₂₇, 16) (dual of [8388600, 8388524, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁷⁶, large, F₂₇, 16) (dual of [large, large−76, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(80, 96, large)-Net over F₂₇ — Digital

Digital (80, 96, large)-net over F₂₇, using

5 times m-reduction [i] based on digital (80, 101, large)-net over F₂₇, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27¹⁰¹, large, F₂₇, 21) (dual of [large, large−101, 22]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(80, 96, large)-Net in Base 27 — Upper bound on s

There is no (80, 96, large)-net in base 27, because

14 times m-reduction [i] would yield (80, 82, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (80, 96, s)-Nets in Base 27

(80, 96, 1049331)-Net over F27 — Constructive and digital

(80, 96, 1050215)-Net in Base 27 — Constructive

(80, 96, large)-Net over F27 — Digital

(80, 96, large)-Net in Base 27 — Upper bound on s

(80, 96, 1049331)-Net over F₂₇ — Constructive and digital

(80, 96, large)-Net over F₂₇ — Digital