MinT - Best Known (109−18, 109, s)-Nets in Base 32

Best Known (109−18, 109, s)-Nets in Base 32

(109−18, 109, 933123)-Net over F₃₂ — Constructive and digital

Digital (91, 109, 933123)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (14, 23, 1056)-net over F₃₂, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 33)-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, 33)-net over F₃₂ (see above)
    3. digital (0, 0, 33)-net over F₃₂ (see above)
    4. digital (0, 0, 33)-net over F₃₂ (see above)
    5. digital (0, 0, 33)-net over F₃₂ (see above)
    6. digital (0, 0, 33)-net over F₃₂ (see above)
    7. digital (0, 0, 33)-net over F₃₂ (see above)
    8. digital (0, 0, 33)-net over F₃₂ (see above)
    9. digital (0, 0, 33)-net over F₃₂ (see above)
    10. digital (0, 0, 33)-net over F₃₂ (see above)
    11. digital (0, 0, 33)-net over F₃₂ (see above)
    12. digital (0, 0, 33)-net over F₃₂ (see above)
    13. digital (0, 0, 33)-net over F₃₂ (see above)
    14. digital (0, 0, 33)-net over F₃₂ (see above)
    15. digital (0, 0, 33)-net over F₃₂ (see above)
    16. digital (0, 0, 33)-net over F₃₂ (see above)
    17. digital (0, 0, 33)-net over F₃₂ (see above)
    18. digital (0, 0, 33)-net over F₃₂ (see above)
    19. digital (0, 0, 33)-net over F₃₂ (see above)
    20. digital (0, 0, 33)-net over F₃₂ (see above)
    21. digital (0, 0, 33)-net over F₃₂ (see above)
    22. digital (0, 0, 33)-net over F₃₂ (see above)
    23. digital (0, 0, 33)-net over F₃₂ (see above)
    24. digital (0, 1, 33)-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]
    25. digital (0, 1, 33)-net over F₃₂ (see above)
    26. digital (0, 1, 33)-net over F₃₂ (see above)
    27. digital (0, 1, 33)-net over F₃₂ (see above)
    28. digital (0, 1, 33)-net over F₃₂ (see above)
    29. digital (0, 2, 33)-net over F₃₂, using
      - kÂ =Â 2 construction [i]
    30. digital (0, 3, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
        the rational function field F₃₂(x) [i]
        Niederreiter sequence [i]
    31. digital (0, 4, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
    32. digital (0, 9, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
2. digital (68, 86, 932067)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁸⁶, 932067, F₃₂, 18, 18) (dual of [(932067, 18), 16777120, 19]-NRT-code), using
    - OA 9-folding and stacking [i] based on linear OA(32⁸⁶, large, F₃₂, 18) (dual of [large, large−86, 19]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(109−18, 109, 940260)-Net in Base 32 — Constructive

(91, 109, 940260)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (17, 26, 8193)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32²⁶, 8193, F₃₂, 9, 9) (dual of [(8193, 9), 73711, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(32²⁶, 32773, F₃₂, 9) (dual of [32773, 32747, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(32²⁶, 32776, F₃₂, 9) (dual of [32776, 32750, 10]-code), using
        constructionÂ X applied to C([0,4]) âŠ‚ C([0,3]) [i] based on
        linear OA(32²⁵, 32769, F₃₂, 9) (dual of [32769, 32744, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769Â | 32⁶âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
        linear OA(32¹⁹, 32769, F₃₂, 7) (dual of [32769, 32750, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769Â | 32⁶âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
        linear OA(32¹, 7, F₃₂, 1) (dual of [7, 6, 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]
2. (65, 83, 932067)-net in base 32, using
  - net defined by OOA [i] based on OOA(32⁸³, 932067, S₃₂, 18, 18), using
    - OA 9-folding and stacking [i] based on OA(32⁸³, large, S₃₂, 18), using
      - discarding parts of the base [i] based on linear OA(64⁶⁹, large, F₆₄, 18) (dual of [large, large−69, 19]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(109−18, 109, large)-Net over F₃₂ — Digital

Digital (91, 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]

(109−18, 109, large)-Net in Base 32 — Upper bound on s

There is no (91, 109, large)-net in base 32, because

16 times m-reduction [i] would yield (91, 93, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (109−18, 109, s)-Nets in Base 32

(109−18, 109, 933123)-Net over F32 — Constructive and digital

(109−18, 109, 940260)-Net in Base 32 — Constructive

(109−18, 109, large)-Net over F32 — Digital

(109−18, 109, large)-Net in Base 32 — Upper bound on s

(109−18, 109, 933123)-Net over F₃₂ — Constructive and digital

(109−18, 109, large)-Net over F₃₂ — Digital