MinT - Best Known (20−4, 20, s)-Nets in Base 32

Best Known (20−4, 20, s)-Nets in Base 32

(20−4, 20, large)-Net over F₃₂ — Constructive and digital

Digital (16, 20, large)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 524290)-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, 524290)-net over F₃₂ (see above)
3. digital (0, 0, 524290)-net over F₃₂ (see above)
4. digital (0, 0, 524290)-net over F₃₂ (see above)
5. digital (0, 0, 524290)-net over F₃₂ (see above)
6. digital (0, 0, 524290)-net over F₃₂ (see above)
7. digital (0, 0, 524290)-net over F₃₂ (see above)
8. digital (0, 0, 524290)-net over F₃₂ (see above)
9. digital (0, 0, 524290)-net over F₃₂ (see above)
10. digital (0, 0, 524290)-net over F₃₂ (see above)
11. digital (0, 0, 524290)-net over F₃₂ (see above)
12. digital (0, 0, 524290)-net over F₃₂ (see above)
13. digital (0, 0, 524290)-net over F₃₂ (see above)
14. digital (0, 0, 524290)-net over F₃₂ (see above)
15. digital (0, 0, 524290)-net over F₃₂ (see above)
16. digital (0, 0, 524290)-net over F₃₂ (see above)
17. digital (0, 0, 524290)-net over F₃₂ (see above)
18. digital (0, 0, 524290)-net over F₃₂ (see above)
19. digital (0, 0, 524290)-net over F₃₂ (see above)
20. digital (0, 0, 524290)-net over F₃₂ (see above)
21. digital (0, 0, 524290)-net over F₃₂ (see above)
22. digital (0, 0, 524290)-net over F₃₂ (see above)
23. digital (0, 0, 524290)-net over F₃₂ (see above)
24. digital (0, 0, 524290)-net over F₃₂ (see above)
25. digital (0, 0, 524290)-net over F₃₂ (see above)
26. digital (0, 0, 524290)-net over F₃₂ (see above)
27. digital (0, 0, 524290)-net over F₃₂ (see above)
28. digital (0, 0, 524290)-net over F₃₂ (see above)
29. digital (0, 1, 524290)-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]
30. digital (0, 1, 524290)-net over F₃₂ (see above)
31. digital (3, 5, 524290)-net over F₃₂, using
  - s-reduction based on digital (3, 5, 1082401)-net over F₃₂, using
    - kÂ =Â 2 construction [i]
32. digital (9, 13, 524290)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32¹³, 524290, F₃₂, 4, 4) (dual of [(524290, 4), 2097147, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(32¹³, 1048580, F₃₂, 4) (dual of [1048580, 1048567, 5]-code), using
      - constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(32¹³, 1048576, F₃₂, 4) (dual of [1048576, 1048563, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(32⁹, 1048576, F₃₂, 3) (dual of [1048576, 1048567, 4]-code or 1048576-cap in PG(8,32)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(32⁰, 4, F₃₂, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(32⁰, s, F₃₂, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(20−4, 20, large)-Net in Base 32 — Upper bound on s

There is no (16, 20, large)-net in base 32, because

2 times m-reduction [i] would yield (16, 18, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (20−4, 20, s)-Nets in Base 32

(20−4, 20, large)-Net over F32 — Constructive and digital

(20−4, 20, large)-Net in Base 32 — Upper bound on s

(20−4, 20, large)-Net over F₃₂ — Constructive and digital