MinT - Best Known (20−8, 20, s)-Nets in Base 32

Best Known (20−8, 20, s)-Nets in Base 32

(20−8, 20, 1056)-Net over F₃₂ — Constructive and digital

Digital (12, 20, 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, 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]
26. digital (0, 1, 33)-net over F₃₂ (see above)
27. digital (0, 1, 33)-net over F₃₂ (see above)
28. digital (0, 1, 33)-net over F₃₂ (see above)
29. digital (0, 2, 33)-net over F₃₂, using
  - kÂ =Â 2 construction [i]
30. digital (0, 2, 33)-net over F₃₂ (see above)
31. digital (0, 4, 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]
32. digital (0, 8, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)

(20−8, 20, 2195)-Net over F₃₂ — Digital

Digital (12, 20, 2195)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32²⁰, 2195, F₃₂, 8) (dual of [2195, 2175, 9]-code), using
- 1163 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 11 times 0, 1, 72 times 0, 1, 305 times 0, 1, 771 times 0) [i] based on linear OA(32¹⁵, 1027, F₃₂, 8) (dual of [1027, 1012, 9]-code), using
  - constructionÂ XX applied to C₁Â =Â C([1022,5]), C₂Â =Â C([0,6]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,5]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([1022,6]) [i] based on
    1. linear OA(32¹³, 1023, F₃₂, 7) (dual of [1023, 1010, 8]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {−1,0,…,5}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
    2. linear OA(32¹³, 1023, F₃₂, 7) (dual of [1023, 1010, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
    3. linear OA(32¹⁵, 1023, F₃₂, 8) (dual of [1023, 1008, 9]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {−1,0,…,6}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
    4. linear OA(32¹¹, 1023, F₃₂, 6) (dual of [1023, 1012, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
    5. linear OA(32⁰, 2, F₃₂, 0) (dual of [2, 2, 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]
    6. linear OA(32⁰, 2, F₃₂, 0) (dual of [2, 2, 1]-code) (see above)

(20−8, 20, 2395745)-Net in Base 32 — Upper bound on s

There is no (12, 20, 2395746)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1 267651 287037 257134 102066 686216 > 32²⁰ [i]

Best Known (20−8, 20, s)-Nets in Base 32

(20−8, 20, 1056)-Net over F32 — Constructive and digital

(20−8, 20, 2195)-Net over F32 — Digital

(20−8, 20, 2395745)-Net in Base 32 — Upper bound on s

(20−8, 20, 1056)-Net over F₃₂ — Constructive and digital

(20−8, 20, 2195)-Net over F₃₂ — Digital