MinT - Best Known (5, 10, s)-Nets in Base 32

Best Known (5, 10, s)-Nets in Base 32

(5, 10, 1056)-Net over F₃₂ — Constructive and digital

Digital (5, 10, 1056)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 33)-net over F₃₂, using
  - s-reduction based on digital (0, 0, s)-net over F₃₂ with arbitrarily large s, using
    - arbitrary set of 32⁰ points [i]
2. digital (0, 0, 33)-net over F₃₂ (see above)
3. digital (0, 0, 33)-net over F₃₂ (see above)
4. digital (0, 0, 33)-net over F₃₂ (see above)
5. digital (0, 0, 33)-net over F₃₂ (see above)
6. digital (0, 0, 33)-net over F₃₂ (see above)
7. digital (0, 0, 33)-net over F₃₂ (see above)
8. digital (0, 0, 33)-net over F₃₂ (see above)
9. digital (0, 0, 33)-net over F₃₂ (see above)
10. digital (0, 0, 33)-net over F₃₂ (see above)
11. digital (0, 0, 33)-net over F₃₂ (see above)
12. digital (0, 0, 33)-net over F₃₂ (see above)
13. digital (0, 0, 33)-net over F₃₂ (see above)
14. digital (0, 0, 33)-net over F₃₂ (see above)
15. digital (0, 0, 33)-net over F₃₂ (see above)
16. digital (0, 0, 33)-net over F₃₂ (see above)
17. digital (0, 0, 33)-net over F₃₂ (see above)
18. digital (0, 0, 33)-net over F₃₂ (see above)
19. digital (0, 0, 33)-net over F₃₂ (see above)
20. digital (0, 0, 33)-net over F₃₂ (see above)
21. digital (0, 0, 33)-net over F₃₂ (see above)
22. digital (0, 0, 33)-net over F₃₂ (see above)
23. digital (0, 0, 33)-net over F₃₂ (see above)
24. digital (0, 0, 33)-net over F₃₂ (see above)
25. digital (0, 0, 33)-net over F₃₂ (see above)
26. digital (0, 0, 33)-net over F₃₂ (see above)
27. digital (0, 0, 33)-net over F₃₂ (see above)
28. digital (0, 1, 33)-net over F₃₂, using
  - s-reduction based on digital (0, 1, s)-net over F₃₂ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
29. digital (0, 1, 33)-net over F₃₂ (see above)
30. digital (0, 1, 33)-net over F₃₂ (see above)
31. digital (0, 2, 33)-net over F₃₂, using
  - kÂ =Â 2 construction [i]
32. digital (0, 5, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
      - the rational function field F₃₂(x) [i]
    - Niederreiter sequence [i]

(5, 10, 1920)-Net over F₃₂ — Digital

Digital (5, 10, 1920)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32¹⁰, 1920, F₃₂, 5) (dual of [1920, 1910, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(32¹⁰, 1986, F₃₂, 5) (dual of [1986, 1976, 6]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(32³, 993, F₃₂, 2) (dual of [993, 990, 3]-code), using
      - discarding factors / shortening the dual code based on linear OA(32³, 1023, F₃₂, 2) (dual of [1023, 1020, 3]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [0,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
    2. linear OA(32⁷, 993, F₃₂, 5) (dual of [993, 986, 6]-code), using
      - a code by Danev and Olsson [i]

(5, 10, 8128)-Net in Base 32 — Constructive

(5, 10, 8128)-net in base 32, using

net defined by OOA [i] based on OOA(32¹⁰, 8128, S₃₂, 5, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(32¹⁰, 16257, S₃₂, 5), using
  - discarding parts of the base [i] based on linear OA(128⁷, 16257, F₁₂₈, 5) (dual of [16257, 16250, 6]-code), using
    - a code by Danev and Olsson [i]

(5, 10, 270599)-Net in Base 32 — Upper bound on s

There is no (5, 10, 270600)-net in base 32, because

1 times m-reduction [i] would yield (5, 9, 270600)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 35 184451 780501 > 32⁹ [i]

Best Known (5, 10, s)-Nets in Base 32

(5, 10, 1056)-Net over F32 — Constructive and digital

(5, 10, 1920)-Net over F32 — Digital

(5, 10, 8128)-Net in Base 32 — Constructive

(5, 10, 270599)-Net in Base 32 — Upper bound on s

(5, 10, 1056)-Net over F₃₂ — Constructive and digital

(5, 10, 1920)-Net over F₃₂ — Digital