MinT - Best Known (41, 41+9, s)-Nets in Base 27

Best Known (41, 41+9, s)-Nets in Base 27

(41, 41+9, 2097916)-Net over F₂₇ — Constructive and digital

Digital (41, 50, 2097916)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (5, 9, 766)-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 (1, 5, 38)-net over F₂₇, using
      - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]
2. digital (32, 41, 2097150)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁴¹, 2097150, F₂₇, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(27⁴¹, 8388601, F₂₇, 9) (dual of [8388601, 8388560, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁴¹, large, F₂₇, 9) (dual of [large, large−41, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(41, 41+9, 2100390)-Net in Base 27 — Constructive

(41, 50, 2100390)-net in base 27, using

27¹ times duplication [i] based on (40, 49, 2100390)-net in base 27, using
- (u,Â u+v)-construction [i] based on
  1. (4, 8, 3240)-net in base 27, using
    - base change [i] based on digital (2, 6, 3240)-net over F₈₁, using
      - net defined by OOA [i] based on linear OOA(81⁶, 3240, F₈₁, 4, 4) (dual of [(3240, 4), 12954, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(81⁶, 6480, F₈₁, 4) (dual of [6480, 6474, 5]-code), using
        1 times truncation [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 (32, 41, 2097150)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27⁴¹, 2097150, F₂₇, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
      - OOA 4-folding and stacking with additional row [i] based on linear OA(27⁴¹, 8388601, F₂₇, 9) (dual of [8388601, 8388560, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁴¹, large, F₂₇, 9) (dual of [large, large−41, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(41, 41+9, large)-Net over F₂₇ — Digital

Digital (41, 50, large)-net over F₂₇, using

t-expansion [i] based on digital (40, 50, large)-net over F₂₇, using
- 1 times m-reduction [i] based on digital (40, 51, large)-net over F₂₇, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] 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]

(41, 41+9, large)-Net in Base 27 — Upper bound on s

There is no (41, 50, large)-net in base 27, because

7 times m-reduction [i] would yield (41, 43, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (41, 41+9, s)-Nets in Base 27

(41, 41+9, 2097916)-Net over F27 — Constructive and digital

(41, 41+9, 2100390)-Net in Base 27 — Constructive

(41, 41+9, large)-Net over F27 — Digital

(41, 41+9, large)-Net in Base 27 — Upper bound on s

(41, 41+9, 2097916)-Net over F₂₇ — Constructive and digital

(41, 41+9, large)-Net over F₂₇ — Digital