MinT - Best Known (55, 66, s)-Nets in Base 32

Best Known (55, 66, s)-Nets in Base 32

(55, 66, 1694648)-Net over F₃₂ — Constructive and digital

Digital (55, 66, 1694648)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (10, 15, 16928)-net over F₃₂, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 529)-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, 529)-net over F₃₂ (see above)
    3. digital (0, 0, 529)-net over F₃₂ (see above)
    4. digital (0, 0, 529)-net over F₃₂ (see above)
    5. digital (0, 0, 529)-net over F₃₂ (see above)
    6. digital (0, 0, 529)-net over F₃₂ (see above)
    7. digital (0, 0, 529)-net over F₃₂ (see above)
    8. digital (0, 0, 529)-net over F₃₂ (see above)
    9. digital (0, 0, 529)-net over F₃₂ (see above)
    10. digital (0, 0, 529)-net over F₃₂ (see above)
    11. digital (0, 0, 529)-net over F₃₂ (see above)
    12. digital (0, 0, 529)-net over F₃₂ (see above)
    13. digital (0, 0, 529)-net over F₃₂ (see above)
    14. digital (0, 0, 529)-net over F₃₂ (see above)
    15. digital (0, 0, 529)-net over F₃₂ (see above)
    16. digital (0, 0, 529)-net over F₃₂ (see above)
    17. digital (0, 0, 529)-net over F₃₂ (see above)
    18. digital (0, 0, 529)-net over F₃₂ (see above)
    19. digital (0, 0, 529)-net over F₃₂ (see above)
    20. digital (0, 0, 529)-net over F₃₂ (see above)
    21. digital (0, 0, 529)-net over F₃₂ (see above)
    22. digital (0, 0, 529)-net over F₃₂ (see above)
    23. digital (0, 0, 529)-net over F₃₂ (see above)
    24. digital (0, 0, 529)-net over F₃₂ (see above)
    25. digital (0, 0, 529)-net over F₃₂ (see above)
    26. digital (0, 0, 529)-net over F₃₂ (see above)
    27. digital (0, 0, 529)-net over F₃₂ (see above)
    28. digital (0, 1, 529)-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]
    29. digital (0, 1, 529)-net over F₃₂ (see above)
    30. digital (0, 1, 529)-net over F₃₂ (see above)
    31. digital (1, 3, 529)-net over F₃₂, using
      - s-reduction based on digital (1, 3, 1057)-net over F₃₂, using
        kÂ =Â 2 construction [i]
    32. digital (4, 9, 529)-net over F₃₂, using
      - (u,Â u+v)-construction [i] based on
        digital (0, 2, 33)-net over F₃₂, using
        kÂ =Â 2 construction [i]
        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 (40, 51, 1677720)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁵¹, 1677720, F₃₂, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(32⁵¹, 8388601, F₃₂, 11) (dual of [8388601, 8388550, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁵¹, large, F₃₂, 11) (dual of [large, large−51, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(55, 66, 1808796)-Net in Base 32 — Constructive

(55, 66, 1808796)-net in base 32, using

base change [i] based on digital (44, 55, 1808796)-net over F₆₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 14, 131076)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64¹⁴, 131076, F₆₄, 5, 5) (dual of [(131076, 5), 655366, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(64¹⁴, 262153, F₆₄, 5) (dual of [262153, 262139, 6]-code), using
        constructionÂ X4 applied to C([0,2]) âŠ‚ C([0,1]) [i] based on
        linear OA(64¹³, 262145, F₆₄, 5) (dual of [262145, 262132, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 64⁶âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
        linear OA(64⁷, 262145, F₆₄, 3) (dual of [262145, 262138, 4]-code or 262145-cap in PG(6,64)), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 64⁶âˆ’1, defining interval IÂ =Â [0,1], and minimum distance dÂ â‰¥ |{−1,0,1}|+1Â =Â 4 (BCH-bound) [i]
        linear OA(64⁷, 8, F₆₄, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,64)), using
        dual of repetition code with length 8 [i]
        linear OA(64¹, 8, F₆₄, 1) (dual of [8, 7, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹, 64, F₆₄, 1) (dual of [64, 63, 2]-code), using
        Reedâ€“Solomon code RS(63,64) [i]
  2. digital (30, 41, 1677720)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁴¹, 1677720, F₆₄, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(64⁴¹, 8388601, F₆₄, 11) (dual of [8388601, 8388560, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴¹, large, F₆₄, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(55, 66, large)-Net over F₃₂ — Digital

Digital (55, 66, large)-net over F₃₂, using

t-expansion [i] based on digital (52, 66, large)-net over F₃₂, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁶⁶, large, F₃₂, 14) (dual of [large, large−66, 15]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(55, 66, large)-Net in Base 32 — Upper bound on s

There is no (55, 66, large)-net in base 32, because

9 times m-reduction [i] would yield (55, 57, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (55, 66, s)-Nets in Base 32

(55, 66, 1694648)-Net over F32 — Constructive and digital

(55, 66, 1808796)-Net in Base 32 — Constructive

(55, 66, large)-Net over F32 — Digital

(55, 66, large)-Net in Base 32 — Upper bound on s

(55, 66, 1694648)-Net over F₃₂ — Constructive and digital

(55, 66, large)-Net over F₃₂ — Digital