MinT - Best Known (61−11, 61, s)-Nets in Base 27

Best Known (61−11, 61, s)-Nets in Base 27

(61−11, 61, 1678476)-Net over F₂₇ — Constructive and digital

Digital (50, 61, 1678476)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (5, 10, 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, 0, 28)-net over F₂₇ (see above)
    22. digital (0, 0, 28)-net over F₂₇ (see above)
    23. 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]
    24. digital (0, 1, 28)-net over F₂₇ (see above)
    25. digital (0, 1, 28)-net over F₂₇ (see above)
    26. digital (0, 2, 28)-net over F₂₇, using
      - kÂ =Â 2 construction [i]
    27. digital (0, 5, 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]
2. digital (40, 51, 1677720)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁵¹, 1677720, F₂₇, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(27⁵¹, 8388601, F₂₇, 11) (dual of [8388601, 8388550, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁵¹, large, F₂₇, 11) (dual of [large, large−51, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(61−11, 61, 1680960)-Net in Base 27 — Constructive

(50, 61, 1680960)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. (5, 10, 3240)-net in base 27, using
  - net defined by OOA [i] based on OOA(27¹⁰, 3240, S₂₇, 5, 5), using
    - OOA 2-folding and stacking with additional row [i] based on OA(27¹⁰, 6481, S₂₇, 5), using
      - discarding parts of the base [i] based on linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (40, 51, 1677720)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁵¹, 1677720, F₂₇, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(27⁵¹, 8388601, F₂₇, 11) (dual of [8388601, 8388550, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁵¹, large, F₂₇, 11) (dual of [large, large−51, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(61−11, 61, large)-Net over F₂₇ — Digital

Digital (50, 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]

(61−11, 61, large)-Net in Base 27 — Upper bound on s

There is no (50, 61, large)-net in base 27, because

9 times m-reduction [i] would yield (50, 52, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (61−11, 61, s)-Nets in Base 27

(61−11, 61, 1678476)-Net over F27 — Constructive and digital

(61−11, 61, 1680960)-Net in Base 27 — Constructive

(61−11, 61, large)-Net over F27 — Digital

(61−11, 61, large)-Net in Base 27 — Upper bound on s

(61−11, 61, 1678476)-Net over F₂₇ — Constructive and digital

(61−11, 61, large)-Net over F₂₇ — Digital