MinT - Best Known (24−9, 24, s)-Nets in Base 32

Best Known (24−9, 24, s)-Nets in Base 32

(24−9, 24, 1067)-Net over F₃₂ — Constructive and digital

Digital (15, 24, 1067)-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, 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]
25. digital (0, 1, 33)-net over F₃₂ (see above)
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, 3, 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]
31. digital (0, 4, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
32. digital (1, 10, 44)-net over F₃₂, using
  - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
      - a function field by SÃ©mirat [i]

(24−9, 24, 3983)-Net over F₃₂ — Digital

Digital (15, 24, 3983)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(24−9, 24, 4096)-Net in Base 32 — Constructive

(15, 24, 4096)-net in base 32, using

net defined by OOA [i] based on OOA(32²⁴, 4096, S₃₂, 9, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(32²⁴, 16385, S₃₂, 9), using
  - discarding factors based on OA(32²⁴, 16386, S₃₂, 9), using
    - discarding parts of the base [i] based on linear OA(128¹⁷, 16386, F₁₂₈, 9) (dual of [16386, 16369, 10]-code), using
      - constructionÂ X applied to C_e(8) âŠ‚ C_e(7) [i] based on
        linear OA(128¹⁷, 16384, F₁₂₈, 9) (dual of [16384, 16367, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(128¹⁵, 16384, F₁₂₈, 8) (dual of [16384, 16369, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(128⁰, 2, F₁₂₈, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(24−9, 24, 4139)-Net in Base 32

(15, 24, 4139)-net in base 32, using

base change [i] based on digital (11, 20, 4139)-net over F₆₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64²⁰, 4139, F₆₄, 9) (dual of [4139, 4119, 10]-code), using
  - 38 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 33 times 0) [i] based on linear OA(64¹⁷, 4098, F₆₄, 9) (dual of [4098, 4081, 10]-code), using
    - constructionÂ X applied to C_e(8) âŠ‚ C_e(7) [i] based on
      1. linear OA(64¹⁷, 4096, F₆₄, 9) (dual of [4096, 4079, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
      2. linear OA(64¹⁵, 4096, F₆₄, 8) (dual of [4096, 4081, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
      3. linear OA(64⁰, 2, F₆₄, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(24−9, 24, large)-Net in Base 32 — Upper bound on s

There is no (15, 24, large)-net in base 32, because

7 times m-reduction [i] would yield (15, 17, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]