MinT - Best Known (95−13, 95, s)-Nets in Base 27

Best Known (95−13, 95, s)-Nets in Base 27

(95−13, 95, 2796956)-Net over F₂₇ — Constructive and digital

Digital (82, 95, 2796956)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 8, 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, 0, 28)-net over F₂₇ (see above)
    24. 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]
    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, 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]
2. digital (20, 26, 1398100)-net over F₂₇, using
  - s-reduction based on 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]
3. digital (48, 61, 1398100)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁶¹, 1398100, F₂₇, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(27⁶¹, 8388601, F₂₇, 13) (dual of [8388601, 8388540, 14]-code), using
      - discarding factors / shortening the dual code 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]

(95−13, 95, large)-Net over F₂₇ — Digital

Digital (82, 95, large)-net over F₂₇, using

t-expansion [i] based on digital (80, 95, large)-net over F₂₇, using
- 6 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]

(95−13, 95, large)-Net in Base 27 — Upper bound on s

There is no (82, 95, large)-net in base 27, because

11 times m-reduction [i] would yield (82, 84, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (95−13, 95, s)-Nets in Base 27

(95−13, 95, 2796956)-Net over F27 — Constructive and digital

(95−13, 95, large)-Net over F27 — Digital

(95−13, 95, large)-Net in Base 27 — Upper bound on s

(95−13, 95, 2796956)-Net over F₂₇ — Constructive and digital

(95−13, 95, large)-Net over F₂₇ — Digital