MinT - Best Known (81−11, 81, s)-Nets in Base 32

Best Known (81−11, 81, s)-Nets in Base 32

(81−11, 81, 6710880)-Net over F₃₂ — Constructive and digital

Digital (70, 81, 6710880)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 209715)-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, 209715)-net over F₃₂ (see above)
3. digital (0, 0, 209715)-net over F₃₂ (see above)
4. digital (0, 0, 209715)-net over F₃₂ (see above)
5. digital (0, 0, 209715)-net over F₃₂ (see above)
6. digital (0, 0, 209715)-net over F₃₂ (see above)
7. digital (0, 0, 209715)-net over F₃₂ (see above)
8. digital (0, 0, 209715)-net over F₃₂ (see above)
9. digital (0, 0, 209715)-net over F₃₂ (see above)
10. digital (0, 0, 209715)-net over F₃₂ (see above)
11. digital (0, 0, 209715)-net over F₃₂ (see above)
12. digital (0, 0, 209715)-net over F₃₂ (see above)
13. digital (0, 0, 209715)-net over F₃₂ (see above)
14. digital (0, 0, 209715)-net over F₃₂ (see above)
15. digital (0, 0, 209715)-net over F₃₂ (see above)
16. digital (0, 0, 209715)-net over F₃₂ (see above)
17. digital (0, 0, 209715)-net over F₃₂ (see above)
18. digital (0, 0, 209715)-net over F₃₂ (see above)
19. digital (0, 0, 209715)-net over F₃₂ (see above)
20. digital (0, 0, 209715)-net over F₃₂ (see above)
21. digital (0, 0, 209715)-net over F₃₂ (see above)
22. digital (0, 1, 209715)-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]
23. digital (0, 1, 209715)-net over F₃₂ (see above)
24. digital (0, 1, 209715)-net over F₃₂ (see above)
25. digital (0, 1, 209715)-net over F₃₂ (see above)
26. digital (0, 1, 209715)-net over F₃₂ (see above)
27. digital (0, 1, 209715)-net over F₃₂ (see above)
28. digital (3, 5, 209715)-net over F₃₂, using
  - s-reduction based on digital (3, 5, 1082401)-net over F₃₂, using
    - kÂ =Â 2 construction [i]
29. digital (3, 5, 209715)-net over F₃₂ (see above)
30. digital (4, 7, 209715)-net over F₃₂, using
  - s-reduction based on digital (4, 7, 1050624)-net over F₃₂, using
    - net defined by OOA [i] based on linear OOA(32⁷, 1050624, F₃₂, 3, 3) (dual of [(1050624, 3), 3151865, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(32⁷, 1050624, F₃₂, 2, 3) (dual of [(1050624, 2), 2101241, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(32⁷, 1050624, F₃₂, 3) (dual of [1050624, 1050617, 4]-code or 1050624-cap in PG(6,32)), using
        product of two caps with tangent hyperplane [i]
31. digital (12, 17, 209715)-net over F₃₂, using
  - s-reduction based on digital (12, 17, 524289)-net over F₃₂, using
    - net defined by OOA [i] based on linear OOA(32¹⁷, 524289, F₃₂, 5, 5) (dual of [(524289, 5), 2621428, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(32¹⁷, 1048579, F₃₂, 5) (dual of [1048579, 1048562, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(32¹⁷, 1048580, F₃₂, 5) (dual of [1048580, 1048563, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(32¹⁷, 1048576, F₃₂, 5) (dual of [1048576, 1048559, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        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(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]
32. digital (30, 41, 209715)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁴¹, 209715, F₃₂, 11, 11) (dual of [(209715, 11), 2306824, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(32⁴¹, 1048576, F₃₂, 11) (dual of [1048576, 1048535, 12]-code), using
      - an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(81−11, 81, large)-Net over F₃₂ — Digital

Digital (70, 81, large)-net over F₃₂, using

t-expansion [i] based on digital (68, 81, large)-net over F₃₂, using
- 5 times m-reduction [i] based on digital (68, 86, large)-net over F₃₂, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(81−11, 81, large)-Net in Base 32 — Upper bound on s

There is no (70, 81, large)-net in base 32, because

9 times m-reduction [i] would yield (70, 72, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (81−11, 81, s)-Nets in Base 32

(81−11, 81, 6710880)-Net over F32 — Constructive and digital

(81−11, 81, large)-Net over F32 — Digital

(81−11, 81, large)-Net in Base 32 — Upper bound on s

(81−11, 81, 6710880)-Net over F₃₂ — Constructive and digital

(81−11, 81, large)-Net over F₃₂ — Digital