MinT - Best Known (85−11, 85, s)-Nets in Base 32

Best Known (85−11, 85, s)-Nets in Base 32

(85−11, 85, 6711008)-Net over F₃₂ — Constructive and digital

Digital (74, 85, 6711008)-net over F₃₂, using

32¹ times duplication [i] based on digital (73, 84, 6711008)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 0, 209719)-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, 209719)-net over F₃₂ (see above)
  3. digital (0, 0, 209719)-net over F₃₂ (see above)
  4. digital (0, 0, 209719)-net over F₃₂ (see above)
  5. digital (0, 0, 209719)-net over F₃₂ (see above)
  6. digital (0, 0, 209719)-net over F₃₂ (see above)
  7. digital (0, 0, 209719)-net over F₃₂ (see above)
  8. digital (0, 0, 209719)-net over F₃₂ (see above)
  9. digital (0, 0, 209719)-net over F₃₂ (see above)
  10. digital (0, 0, 209719)-net over F₃₂ (see above)
  11. digital (0, 0, 209719)-net over F₃₂ (see above)
  12. digital (0, 0, 209719)-net over F₃₂ (see above)
  13. digital (0, 0, 209719)-net over F₃₂ (see above)
  14. digital (0, 0, 209719)-net over F₃₂ (see above)
  15. digital (0, 0, 209719)-net over F₃₂ (see above)
  16. digital (0, 0, 209719)-net over F₃₂ (see above)
  17. digital (0, 0, 209719)-net over F₃₂ (see above)
  18. digital (0, 0, 209719)-net over F₃₂ (see above)
  19. digital (0, 0, 209719)-net over F₃₂ (see above)
  20. digital (0, 0, 209719)-net over F₃₂ (see above)
  21. digital (0, 0, 209719)-net over F₃₂ (see above)
  22. digital (0, 1, 209719)-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]
  23. digital (0, 1, 209719)-net over F₃₂ (see above)
  24. digital (0, 1, 209719)-net over F₃₂ (see above)
  25. digital (0, 1, 209719)-net over F₃₂ (see above)
  26. digital (0, 1, 209719)-net over F₃₂ (see above)
  27. digital (0, 1, 209719)-net over F₃₂ (see above)
  28. digital (3, 5, 209719)-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, 209719)-net over F₃₂ (see above)
  30. digital (4, 7, 209719)-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, 209719)-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 (33, 44, 209719)-net over F₃₂, using
    - net defined by OOA [i] based on linear OOA(32⁴⁴, 209719, F₃₂, 11, 11) (dual of [(209719, 11), 2306865, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(32⁴⁴, 1048596, F₃₂, 11) (dual of [1048596, 1048552, 12]-code), using
        constructionÂ X applied to C([0,5]) âŠ‚ C([0,3]) [i] based on
        linear OA(32⁴¹, 1048577, F₃₂, 11) (dual of [1048577, 1048536, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 32⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]
        linear OA(32²⁵, 1048577, F₃₂, 7) (dual of [1048577, 1048552, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 32⁸âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
        linear OA(32³, 19, F₃₂, 3) (dual of [19, 16, 4]-code or 19-arc in PG(2,32) or 19-cap in PG(2,32)), using
        discarding factors / shortening the dual code based on linear OA(32³, 32, F₃₂, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
        Reedâ€“Solomon code RS(29,32) [i]

(85−11, 85, large)-Net over F₃₂ — Digital

Digital (74, 85, large)-net over F₃₂, using

t-expansion [i] based on digital (72, 85, large)-net over F₃₂, using
- 6 times m-reduction [i] based on digital (72, 91, large)-net over F₃₂, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(85−11, 85, large)-Net in Base 32 — Upper bound on s

There is no (74, 85, large)-net in base 32, because

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

Best Known (85−11, 85, s)-Nets in Base 32

(85−11, 85, 6711008)-Net over F32 — Constructive and digital

(85−11, 85, large)-Net over F32 — Digital

(85−11, 85, large)-Net in Base 32 — Upper bound on s

(85−11, 85, 6711008)-Net over F₃₂ — Constructive and digital

(85−11, 85, large)-Net over F₃₂ — Digital