MinT - Best Known (109−13, 109, s)-Nets in Base 32

Best Known (109−13, 109, s)-Nets in Base 32

(109−13, 109, 5592448)-Net over F₃₂ — Constructive and digital

Digital (96, 109, 5592448)-net over F₃₂, using

32¹ times duplication [i] based on digital (95, 108, 5592448)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 0, 174764)-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, 174764)-net over F₃₂ (see above)
  3. digital (0, 0, 174764)-net over F₃₂ (see above)
  4. digital (0, 0, 174764)-net over F₃₂ (see above)
  5. digital (0, 0, 174764)-net over F₃₂ (see above)
  6. digital (0, 0, 174764)-net over F₃₂ (see above)
  7. digital (0, 0, 174764)-net over F₃₂ (see above)
  8. digital (0, 0, 174764)-net over F₃₂ (see above)
  9. digital (0, 0, 174764)-net over F₃₂ (see above)
  10. digital (0, 0, 174764)-net over F₃₂ (see above)
  11. digital (0, 0, 174764)-net over F₃₂ (see above)
  12. digital (0, 0, 174764)-net over F₃₂ (see above)
  13. digital (0, 0, 174764)-net over F₃₂ (see above)
  14. digital (0, 0, 174764)-net over F₃₂ (see above)
  15. digital (0, 0, 174764)-net over F₃₂ (see above)
  16. digital (0, 0, 174764)-net over F₃₂ (see above)
  17. digital (0, 0, 174764)-net over F₃₂ (see above)
  18. digital (0, 0, 174764)-net over F₃₂ (see above)
  19. digital (0, 0, 174764)-net over F₃₂ (see above)
  20. digital (0, 1, 174764)-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]
  21. digital (0, 1, 174764)-net over F₃₂ (see above)
  22. digital (0, 1, 174764)-net over F₃₂ (see above)
  23. digital (0, 1, 174764)-net over F₃₂ (see above)
  24. digital (0, 1, 174764)-net over F₃₂ (see above)
  25. digital (0, 1, 174764)-net over F₃₂ (see above)
  26. digital (0, 1, 174764)-net over F₃₂ (see above)
  27. digital (3, 5, 174764)-net over F₃₂, using
    - s-reduction based on digital (3, 5, 1082401)-net over F₃₂, using
      - kÂ =Â 2 construction [i]
  28. digital (3, 5, 174764)-net over F₃₂ (see above)
  29. digital (4, 7, 174764)-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]
  30. digital (9, 13, 174764)-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]
  31. digital (15, 21, 174764)-net over F₃₂, using
    - s-reduction based on digital (15, 21, 349526)-net over F₃₂, using
      - net defined by OOA [i] based on linear OOA(32²¹, 349526, F₃₂, 6, 6) (dual of [(349526, 6), 2097135, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(32²¹, 1048578, F₃₂, 6) (dual of [1048578, 1048557, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(32²¹, 1048580, F₃₂, 6) (dual of [1048580, 1048559, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(32²¹, 1048576, F₃₂, 6) (dual of [1048576, 1048555, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        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(32⁰, 4, F₃₂, 0) (dual of [4, 4, 1]-code) (see above)
  32. digital (37, 50, 174764)-net over F₃₂, using
    - net defined by OOA [i] based on linear OOA(32⁵⁰, 174764, F₃₂, 13, 13) (dual of [(174764, 13), 2271882, 14]-NRT-code), using
      - OOA 6-folding and stacking with additional row [i] based on linear OA(32⁵⁰, 1048585, F₃₂, 13) (dual of [1048585, 1048535, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(32⁵⁰, 1048586, F₃₂, 13) (dual of [1048586, 1048536, 14]-code), using
        constructionÂ X applied to C([0,6]) âŠ‚ C([0,5]) [i] based on
        linear OA(32⁴⁹, 1048577, F₃₂, 13) (dual of [1048577, 1048528, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 32⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        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(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]

(109−13, 109, large)-Net over F₃₂ — Digital

Digital (96, 109, large)-net over F₃₂, using

t-expansion [i] based on digital (87, 109, large)-net over F₃₂, using
- 1 times m-reduction [i] based on digital (87, 110, large)-net over F₃₂, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(109−13, 109, large)-Net in Base 32 — Upper bound on s

There is no (96, 109, large)-net in base 32, because

11 times m-reduction [i] would yield (96, 98, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (109−13, 109, s)-Nets in Base 32

(109−13, 109, 5592448)-Net over F32 — Constructive and digital

(109−13, 109, large)-Net over F32 — Digital

(109−13, 109, large)-Net in Base 32 — Upper bound on s

(109−13, 109, 5592448)-Net over F₃₂ — Constructive and digital

(109−13, 109, large)-Net over F₃₂ — Digital