MinT - Best Known (84−10, 84, s)-Nets in Base 32

Best Known (84−10, 84, s)-Nets in Base 32

(84−10, 84, 6726816)-Net over F₃₂ — Constructive and digital

Digital (74, 84, 6726816)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 210213)-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, 210213)-net over F₃₂ (see above)
3. digital (0, 0, 210213)-net over F₃₂ (see above)
4. digital (0, 0, 210213)-net over F₃₂ (see above)
5. digital (0, 0, 210213)-net over F₃₂ (see above)
6. digital (0, 0, 210213)-net over F₃₂ (see above)
7. digital (0, 0, 210213)-net over F₃₂ (see above)
8. digital (0, 0, 210213)-net over F₃₂ (see above)
9. digital (0, 0, 210213)-net over F₃₂ (see above)
10. digital (0, 0, 210213)-net over F₃₂ (see above)
11. digital (0, 0, 210213)-net over F₃₂ (see above)
12. digital (0, 0, 210213)-net over F₃₂ (see above)
13. digital (0, 0, 210213)-net over F₃₂ (see above)
14. digital (0, 0, 210213)-net over F₃₂ (see above)
15. digital (0, 0, 210213)-net over F₃₂ (see above)
16. digital (0, 0, 210213)-net over F₃₂ (see above)
17. digital (0, 0, 210213)-net over F₃₂ (see above)
18. digital (0, 0, 210213)-net over F₃₂ (see above)
19. digital (0, 0, 210213)-net over F₃₂ (see above)
20. digital (0, 0, 210213)-net over F₃₂ (see above)
21. digital (0, 0, 210213)-net over F₃₂ (see above)
22. digital (0, 0, 210213)-net over F₃₂ (see above)
23. digital (0, 1, 210213)-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]
24. digital (0, 1, 210213)-net over F₃₂ (see above)
25. digital (0, 1, 210213)-net over F₃₂ (see above)
26. digital (0, 1, 210213)-net over F₃₂ (see above)
27. digital (0, 1, 210213)-net over F₃₂ (see above)
28. digital (3, 5, 210213)-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, 210213)-net over F₃₂ (see above)
30. digital (4, 7, 210213)-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, 210213)-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 (35, 45, 210213)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (2, 7, 496)-net over F₃₂, using
      - net defined by OOA [i] based on linear OOA(32⁷, 496, F₃₂, 5, 5) (dual of [(496, 5), 2473, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(32⁷, 993, F₃₂, 5) (dual of [993, 986, 6]-code), using
        a code by Danev and Olsson [i]
    2. digital (28, 38, 209717)-net over F₃₂, using
      - net defined by OOA [i] based on linear OOA(32³⁸, 209717, F₃₂, 10, 10) (dual of [(209717, 10), 2097132, 11]-NRT-code), using
        OA 5-folding and stacking [i] based on linear OA(32³⁸, 1048585, F₃₂, 10) (dual of [1048585, 1048547, 11]-code), using
        constructionÂ X applied to C_e(9) âŠ‚ C_e(7) [i] based on
        linear OA(32³⁷, 1048576, F₃₂, 10) (dual of [1048576, 1048539, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        linear OA(32²⁹, 1048576, F₃₂, 8) (dual of [1048576, 1048547, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(32¹, 9, F₃₂, 1) (dual of [9, 8, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(32¹, s, F₃₂, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(84−10, 84, large)-Net in Base 32 — Constructive

(74, 84, large)-net in base 32, using

base change [i] based on digital (60, 70, large)-net over F₆₄, using
- 64¹ times duplication [i] based on digital (59, 69, 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, 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]
    56. digital (0, 1, 131073)-net over F₆₄ (see above)
    57. digital (0, 1, 131073)-net over F₆₄ (see above)
    58. digital (0, 1, 131073)-net over F₆₄ (see above)
    59. digital (0, 1, 131073)-net over F₆₄ (see above)
    60. 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]
    61. digital (2, 4, 131073)-net over F₆₄ (see above)
    62. digital (3, 6, 131073)-net over F₆₄, using
      - s-reduction based on digital (3, 6, 270402)-net over F₆₄, using
        net defined by OOA [i] based on linear OOA(64⁶, 270402, F₆₄, 3, 3) (dual of [(270402, 3), 811200, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(64⁶, 270402, F₆₄, 2, 3) (dual of [(270402, 2), 540798, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(64⁶, 270402, F₆₄, 3) (dual of [270402, 270396, 4]-code or 270402-cap in PG(5,64)), using
        large caps in PG(5,Â b) [i]
    63. 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]
    64. digital (27, 37, 1677720)-net over F₆₄, using
      - net defined by OOA [i] based on linear OOA(64³⁷, 1677720, F₆₄, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
        OA 5-folding and stacking [i] based on linear OA(64³⁷, 8388600, F₆₄, 10) (dual of [8388600, 8388563, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(64³⁷, large, F₆₄, 10) (dual of [large, large−37, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(84−10, 84, large)-Net over F₃₂ — Digital

Digital (74, 84, large)-net over F₃₂, using

t-expansion [i] based on digital (72, 84, large)-net over F₃₂, using
- 7 times m-reduction [i] based on digital (72, 91, large)-net over F₃₂, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(84−10, 84, large)-Net in Base 32 — Upper bound on s

There is no (74, 84, large)-net in base 32, because

8 times m-reduction [i] would yield (74, 76, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (84−10, 84, s)-Nets in Base 32

(84−10, 84, 6726816)-Net over F32 — Constructive and digital

(84−10, 84, large)-Net in Base 32 — Constructive

(84−10, 84, large)-Net over F32 — Digital

(84−10, 84, large)-Net in Base 32 — Upper bound on s

(84−10, 84, 6726816)-Net over F₃₂ — Constructive and digital

(84−10, 84, large)-Net over F₃₂ — Digital