MinT - Best Known (19, 24, s)-Nets in Base 32

Best Known (19, 24, s)-Nets in Base 32

(19, 24, 4195358)-Net over F₃₂ — Constructive and digital

Digital (19, 24, 4195358)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 3, 1057)-net over F₃₂, using
  - kÂ =Â 2 construction [i]
2. digital (16, 21, 4194301)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32²¹, 4194301, F₃₂, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OA(32²¹, large, F₃₂, 5) (dual of [large, large−21, 6]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]

(19, 24, large)-Net in Base 32 — Constructive

(19, 24, large)-net in base 32, using

base change [i] based on digital (15, 20, large)-net over F₆₄, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 0, 131073)-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, 131073)-net over F₆₄ (see above)
  3. digital (0, 0, 131073)-net over F₆₄ (see above)
  4. digital (0, 0, 131073)-net over F₆₄ (see above)
  5. digital (0, 0, 131073)-net over F₆₄ (see above)
  6. digital (0, 0, 131073)-net over F₆₄ (see above)
  7. digital (0, 0, 131073)-net over F₆₄ (see above)
  8. digital (0, 0, 131073)-net over F₆₄ (see above)
  9. digital (0, 0, 131073)-net over F₆₄ (see above)
  10. digital (0, 0, 131073)-net over F₆₄ (see above)
  11. digital (0, 0, 131073)-net over F₆₄ (see above)
  12. digital (0, 0, 131073)-net over F₆₄ (see above)
  13. digital (0, 0, 131073)-net over F₆₄ (see above)
  14. digital (0, 0, 131073)-net over F₆₄ (see above)
  15. digital (0, 0, 131073)-net over F₆₄ (see above)
  16. digital (0, 0, 131073)-net over F₆₄ (see above)
  17. digital (0, 0, 131073)-net over F₆₄ (see above)
  18. digital (0, 0, 131073)-net over F₆₄ (see above)
  19. digital (0, 0, 131073)-net over F₆₄ (see above)
  20. digital (0, 0, 131073)-net over F₆₄ (see above)
  21. digital (0, 0, 131073)-net over F₆₄ (see above)
  22. digital (0, 0, 131073)-net over F₆₄ (see above)
  23. digital (0, 0, 131073)-net over F₆₄ (see above)
  24. digital (0, 0, 131073)-net over F₆₄ (see above)
  25. digital (0, 0, 131073)-net over F₆₄ (see above)
  26. digital (0, 0, 131073)-net over F₆₄ (see above)
  27. digital (0, 0, 131073)-net over F₆₄ (see above)
  28. digital (0, 0, 131073)-net over F₆₄ (see above)
  29. digital (0, 0, 131073)-net over F₆₄ (see above)
  30. digital (0, 0, 131073)-net over F₆₄ (see above)
  31. digital (0, 0, 131073)-net over F₆₄ (see above)
  32. digital (0, 0, 131073)-net over F₆₄ (see above)
  33. digital (0, 0, 131073)-net over F₆₄ (see above)
  34. digital (0, 0, 131073)-net over F₆₄ (see above)
  35. digital (0, 0, 131073)-net over F₆₄ (see above)
  36. digital (0, 0, 131073)-net over F₆₄ (see above)
  37. digital (0, 0, 131073)-net over F₆₄ (see above)
  38. digital (0, 0, 131073)-net over F₆₄ (see above)
  39. digital (0, 0, 131073)-net over F₆₄ (see above)
  40. digital (0, 0, 131073)-net over F₆₄ (see above)
  41. digital (0, 0, 131073)-net over F₆₄ (see above)
  42. digital (0, 0, 131073)-net over F₆₄ (see above)
  43. digital (0, 0, 131073)-net over F₆₄ (see above)
  44. digital (0, 0, 131073)-net over F₆₄ (see above)
  45. digital (0, 0, 131073)-net over F₆₄ (see above)
  46. digital (0, 0, 131073)-net over F₆₄ (see above)
  47. digital (0, 0, 131073)-net over F₆₄ (see above)
  48. digital (0, 0, 131073)-net over F₆₄ (see above)
  49. digital (0, 0, 131073)-net over F₆₄ (see above)
  50. digital (0, 0, 131073)-net over F₆₄ (see above)
  51. digital (0, 0, 131073)-net over F₆₄ (see above)
  52. digital (0, 0, 131073)-net over F₆₄ (see above)
  53. digital (0, 0, 131073)-net over F₆₄ (see above)
  54. digital (0, 0, 131073)-net over F₆₄ (see above)
  55. digital (0, 0, 131073)-net over F₆₄ (see above)
  56. digital (0, 0, 131073)-net over F₆₄ (see above)
  57. digital (0, 0, 131073)-net over F₆₄ (see above)
  58. digital (0, 0, 131073)-net over F₆₄ (see above)
  59. digital (0, 0, 131073)-net over F₆₄ (see above)
  60. digital (0, 1, 131073)-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]
  61. digital (0, 1, 131073)-net over F₆₄ (see above)
  62. digital (0, 1, 131073)-net over F₆₄ (see above)
  63. digital (2, 4, 131073)-net over F₆₄, using
    - s-reduction based on digital (2, 4, 266305)-net over F₆₄, using
      - kÂ =Â 2 construction [i]
  64. digital (8, 13, 131073)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64¹³, 131073, F₆₄, 5, 5) (dual of [(131073, 5), 655352, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(64¹³, 262147, F₆₄, 5) (dual of [262147, 262134, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(64¹³, 262144, F₆₄, 5) (dual of [262144, 262131, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(64¹⁰, 262144, F₆₄, 4) (dual of [262144, 262134, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 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]

(19, 24, large)-Net over F₃₂ — Digital

Digital (19, 24, large)-net over F₃₂, using

32³ times duplication [i] based on digital (16, 21, large)-net over F₃₂, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32²¹, large, F₃₂, 5) (dual of [large, large−21, 6]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]

(19, 24, large)-Net in Base 32 — Upper bound on s

There is no (19, 24, large)-net in base 32, because

3 times m-reduction [i] would yield (19, 21, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (19, 24, s)-Nets in Base 32

(19, 24, 4195358)-Net over F32 — Constructive and digital

(19, 24, large)-Net in Base 32 — Constructive

(19, 24, large)-Net over F32 — Digital

(19, 24, large)-Net in Base 32 — Upper bound on s

(19, 24, 4195358)-Net over F₃₂ — Constructive and digital

(19, 24, large)-Net over F₃₂ — Digital