MinT - Best Known (10, 10+7, s)-Nets in Base 64

Best Known (10, 10+7, s)-Nets in Base 64

(10, 10+7, 4175)-Net over F₆₄ — Constructive and digital

Digital (10, 17, 4175)-net over F₆₄, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 65)-net over F₆₄, using
  - s-reduction based on digital (0, 0, s)-net over F₆₄ with arbitrarily large s, using
    - arbitrary set of 64⁰ points [i]
2. digital (0, 0, 65)-net over F₆₄ (see above)
3. digital (0, 0, 65)-net over F₆₄ (see above)
4. digital (0, 0, 65)-net over F₆₄ (see above)
5. digital (0, 0, 65)-net over F₆₄ (see above)
6. digital (0, 0, 65)-net over F₆₄ (see above)
7. digital (0, 0, 65)-net over F₆₄ (see above)
8. digital (0, 0, 65)-net over F₆₄ (see above)
9. digital (0, 0, 65)-net over F₆₄ (see above)
10. digital (0, 0, 65)-net over F₆₄ (see above)
11. digital (0, 0, 65)-net over F₆₄ (see above)
12. digital (0, 0, 65)-net over F₆₄ (see above)
13. digital (0, 0, 65)-net over F₆₄ (see above)
14. digital (0, 0, 65)-net over F₆₄ (see above)
15. digital (0, 0, 65)-net over F₆₄ (see above)
16. digital (0, 0, 65)-net over F₆₄ (see above)
17. digital (0, 0, 65)-net over F₆₄ (see above)
18. digital (0, 0, 65)-net over F₆₄ (see above)
19. digital (0, 0, 65)-net over F₆₄ (see above)
20. digital (0, 0, 65)-net over F₆₄ (see above)
21. digital (0, 0, 65)-net over F₆₄ (see above)
22. digital (0, 0, 65)-net over F₆₄ (see above)
23. digital (0, 0, 65)-net over F₆₄ (see above)
24. digital (0, 0, 65)-net over F₆₄ (see above)
25. digital (0, 0, 65)-net over F₆₄ (see above)
26. digital (0, 0, 65)-net over F₆₄ (see above)
27. digital (0, 0, 65)-net over F₆₄ (see above)
28. digital (0, 0, 65)-net over F₆₄ (see above)
29. digital (0, 0, 65)-net over F₆₄ (see above)
30. digital (0, 0, 65)-net over F₆₄ (see above)
31. digital (0, 0, 65)-net over F₆₄ (see above)
32. digital (0, 0, 65)-net over F₆₄ (see above)
33. digital (0, 0, 65)-net over F₆₄ (see above)
34. digital (0, 0, 65)-net over F₆₄ (see above)
35. digital (0, 0, 65)-net over F₆₄ (see above)
36. digital (0, 0, 65)-net over F₆₄ (see above)
37. digital (0, 0, 65)-net over F₆₄ (see above)
38. digital (0, 0, 65)-net over F₆₄ (see above)
39. digital (0, 0, 65)-net over F₆₄ (see above)
40. digital (0, 0, 65)-net over F₆₄ (see above)
41. digital (0, 0, 65)-net over F₆₄ (see above)
42. digital (0, 0, 65)-net over F₆₄ (see above)
43. digital (0, 0, 65)-net over F₆₄ (see above)
44. digital (0, 0, 65)-net over F₆₄ (see above)
45. digital (0, 0, 65)-net over F₆₄ (see above)
46. digital (0, 0, 65)-net over F₆₄ (see above)
47. digital (0, 0, 65)-net over F₆₄ (see above)
48. digital (0, 0, 65)-net over F₆₄ (see above)
49. digital (0, 0, 65)-net over F₆₄ (see above)
50. digital (0, 0, 65)-net over F₆₄ (see above)
51. digital (0, 0, 65)-net over F₆₄ (see above)
52. digital (0, 0, 65)-net over F₆₄ (see above)
53. digital (0, 0, 65)-net over F₆₄ (see above)
54. digital (0, 0, 65)-net over F₆₄ (see above)
55. digital (0, 0, 65)-net over F₆₄ (see above)
56. digital (0, 0, 65)-net over F₆₄ (see above)
57. digital (0, 0, 65)-net over F₆₄ (see above)
58. digital (0, 1, 65)-net over F₆₄, using
  - s-reduction based on digital (0, 1, s)-net over F₆₄ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
59. digital (0, 1, 65)-net over F₆₄ (see above)
60. digital (0, 1, 65)-net over F₆₄ (see above)
61. digital (0, 1, 65)-net over F₆₄ (see above)
62. digital (0, 2, 65)-net over F₆₄, using
  - kÂ =Â 2 construction [i]
63. digital (0, 3, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]
64. digital (1, 8, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(10, 10+7, 5463)-Net in Base 64 — Constructive

(10, 17, 5463)-net in base 64, using

net defined by OOA [i] based on OOA(64¹⁷, 5463, S₆₄, 7, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(64¹⁷, 16390, S₆₄, 7), using
  - discarding parts of the base [i] based on linear OA(128¹⁴, 16390, F₁₂₈, 7) (dual of [16390, 16376, 8]-code), using
    - constructionÂ X applied to C([0,3]) âŠ‚ C([0,2]) [i] based on
      1. linear OA(128¹³, 16385, F₁₂₈, 7) (dual of [16385, 16372, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 128⁴âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
      2. linear OA(128⁹, 16385, F₁₂₈, 5) (dual of [16385, 16376, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 128⁴âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
      3. linear OA(128¹, 5, F₁₂₈, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(128¹, s, F₁₂₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(10, 10+7, 8195)-Net over F₆₄ — Digital

Digital (10, 17, 8195)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64¹⁷, 8195, F₆₄, 7) (dual of [8195, 8178, 8]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(64⁴, 4097, F₆₄, 3) (dual of [4097, 4093, 4]-code or 4097-cap in PG(3,64)), using
    - ovoid in PG(3,Â 64) [i]
  2. linear OA(64¹³, 4098, F₆₄, 7) (dual of [4098, 4085, 8]-code), using
    - constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
      1. linear OA(64¹³, 4096, F₆₄, 7) (dual of [4096, 4083, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
      2. linear OA(64¹¹, 4096, F₆₄, 6) (dual of [4096, 4085, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [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]

(10, 10+7, large)-Net in Base 64 — Upper bound on s

There is no (10, 17, large)-net in base 64, because

5 times m-reduction [i] would yield (10, 12, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (10, 10+7, s)-Nets in Base 64

(10, 10+7, 4175)-Net over F64 — Constructive and digital

(10, 10+7, 5463)-Net in Base 64 — Constructive

(10, 10+7, 8195)-Net over F64 — Digital

(10, 10+7, large)-Net in Base 64 — Upper bound on s

(10, 10+7, 4175)-Net over F₆₄ — Constructive and digital

(10, 10+7, 8195)-Net over F₆₄ — Digital