MinT - Best Known (19, 19+4, s)-Nets in Base 27

Best Known (19, 19+4, s)-Nets in Base 27

(19, 19+4, large)-Net over F₂₇ — Constructive and digital

Digital (19, 23, large)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 551881)-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, 551881)-net over F₂₇ (see above)
3. digital (0, 0, 551881)-net over F₂₇ (see above)
4. digital (0, 0, 551881)-net over F₂₇ (see above)
5. digital (0, 0, 551881)-net over F₂₇ (see above)
6. digital (0, 0, 551881)-net over F₂₇ (see above)
7. digital (0, 0, 551881)-net over F₂₇ (see above)
8. digital (0, 0, 551881)-net over F₂₇ (see above)
9. digital (0, 0, 551881)-net over F₂₇ (see above)
10. digital (0, 0, 551881)-net over F₂₇ (see above)
11. digital (0, 0, 551881)-net over F₂₇ (see above)
12. digital (0, 0, 551881)-net over F₂₇ (see above)
13. digital (0, 0, 551881)-net over F₂₇ (see above)
14. digital (0, 0, 551881)-net over F₂₇ (see above)
15. digital (0, 0, 551881)-net over F₂₇ (see above)
16. digital (0, 0, 551881)-net over F₂₇ (see above)
17. digital (0, 0, 551881)-net over F₂₇ (see above)
18. digital (0, 0, 551881)-net over F₂₇ (see above)
19. digital (0, 0, 551881)-net over F₂₇ (see above)
20. digital (0, 0, 551881)-net over F₂₇ (see above)
21. digital (0, 0, 551881)-net over F₂₇ (see above)
22. digital (0, 0, 551881)-net over F₂₇ (see above)
23. digital (0, 0, 551881)-net over F₂₇ (see above)
24. digital (0, 1, 551881)-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, 551881)-net over F₂₇ (see above)
26. digital (3, 5, 551881)-net over F₂₇, using
  - kÂ =Â 2 construction [i]
27. digital (12, 16, 4194301)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27¹⁶, 4194301, F₂₇, 4, 4) (dual of [(4194301, 4), 16777188, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(27¹⁶, 8388602, F₂₇, 4) (dual of [8388602, 8388586, 5]-code), using
      - discarding factors / shortening the dual code based on linear OA(27¹⁶, large, F₂₇, 4) (dual of [large, large−16, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

(19, 19+4, large)-Net in Base 27 — Upper bound on s

There is no (19, 23, large)-net in base 27, because

2 times m-reduction [i] would yield (19, 21, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (19, 19+4, s)-Nets in Base 27

(19, 19+4, large)-Net over F27 — Constructive and digital

(19, 19+4, large)-Net in Base 27 — Upper bound on s

(19, 19+4, large)-Net over F₂₇ — Constructive and digital