MinT - Best Known (55−6, 55, s)-Nets in Base 32

Best Known (55−6, 55, s)-Nets in Base 32

(55−6, 55, large)-Net over F₃₂ — Constructive and digital

Digital (49, 55, large)-net over F₃₂, using

t-expansion [i] based on digital (48, 55, large)-net over F₃₂, using
- 1 times m-reduction [i] based on digital (48, 56, large)-net over F₃₂, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 262145)-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, 262145)-net over F₃₂ (see above)
    3. digital (0, 0, 262145)-net over F₃₂ (see above)
    4. digital (0, 0, 262145)-net over F₃₂ (see above)
    5. digital (0, 0, 262145)-net over F₃₂ (see above)
    6. digital (0, 0, 262145)-net over F₃₂ (see above)
    7. digital (0, 0, 262145)-net over F₃₂ (see above)
    8. digital (0, 0, 262145)-net over F₃₂ (see above)
    9. digital (0, 0, 262145)-net over F₃₂ (see above)
    10. digital (0, 0, 262145)-net over F₃₂ (see above)
    11. digital (0, 0, 262145)-net over F₃₂ (see above)
    12. digital (0, 0, 262145)-net over F₃₂ (see above)
    13. digital (0, 0, 262145)-net over F₃₂ (see above)
    14. digital (0, 0, 262145)-net over F₃₂ (see above)
    15. digital (0, 0, 262145)-net over F₃₂ (see above)
    16. digital (0, 0, 262145)-net over F₃₂ (see above)
    17. digital (0, 0, 262145)-net over F₃₂ (see above)
    18. digital (0, 0, 262145)-net over F₃₂ (see above)
    19. digital (0, 0, 262145)-net over F₃₂ (see above)
    20. digital (0, 0, 262145)-net over F₃₂ (see above)
    21. digital (0, 0, 262145)-net over F₃₂ (see above)
    22. digital (0, 0, 262145)-net over F₃₂ (see above)
    23. digital (0, 0, 262145)-net over F₃₂ (see above)
    24. digital (0, 0, 262145)-net over F₃₂ (see above)
    25. digital (0, 1, 262145)-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]
    26. digital (0, 1, 262145)-net over F₃₂ (see above)
    27. digital (0, 1, 262145)-net over F₃₂ (see above)
    28. digital (0, 1, 262145)-net over F₃₂ (see above)
    29. digital (3, 5, 262145)-net over F₃₂, using
      - s-reduction based on digital (3, 5, 1082401)-net over F₃₂, using
        kÂ =Â 2 construction [i]
    30. digital (3, 5, 262145)-net over F₃₂ (see above)
    31. digital (9, 13, 262145)-net over F₃₂, using
      - s-reduction based on digital (9, 13, 524290)-net over F₃₂, using
        net defined by OOA [i] based on linear OOA(32¹³, 524290, F₃₂, 4, 4) (dual of [(524290, 4), 2097147, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(32¹³, 1048580, F₃₂, 4) (dual of [1048580, 1048567, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        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(32⁹, 1048576, F₃₂, 3) (dual of [1048576, 1048567, 4]-code or 1048576-cap in PG(8,32)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [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 (21, 29, 262145)-net over F₃₂, using
      - net defined by OOA [i] based on linear OOA(32²⁹, 262145, F₃₂, 8, 8) (dual of [(262145, 8), 2097131, 9]-NRT-code), using
        OA 4-folding and stacking [i] based on linear OA(32²⁹, 1048580, F₃₂, 8) (dual of [1048580, 1048551, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        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(32²⁵, 1048576, F₃₂, 7) (dual of [1048576, 1048551, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(32⁰, 4, F₃₂, 0) (dual of [4, 4, 1]-code) (see above)

(55−6, 55, large)-Net in Base 32 — Upper bound on s

There is no (49, 55, large)-net in base 32, because

4 times m-reduction [i] would yield (49, 51, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (55−6, 55, s)-Nets in Base 32

(55−6, 55, large)-Net over F32 — Constructive and digital

(55−6, 55, large)-Net in Base 32 — Upper bound on s

(55−6, 55, large)-Net over F₃₂ — Constructive and digital