MinT - Best Known (59, 71, s)-Nets in Base 32

Best Known (59, 71, s)-Nets in Base 32

(59, 71, 1399167)-Net over F₃₂ — Constructive and digital

Digital (59, 71, 1399167)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (9, 15, 1067)-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, 0, 33)-net over F₃₂ (see above)
    25. digital (0, 0, 33)-net over F₃₂ (see above)
    26. digital (0, 0, 33)-net over F₃₂ (see above)
    27. 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]
    28. digital (0, 1, 33)-net over F₃₂ (see above)
    29. digital (0, 1, 33)-net over F₃₂ (see above)
    30. digital (0, 2, 33)-net over F₃₂, using
      - kÂ =Â 2 construction [i]
    31. 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]
    32. digital (1, 7, 44)-net over F₃₂, using
      - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
        a function field by SÃ©mirat [i]
2. digital (44, 56, 1398100)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁵⁶, 1398100, F₃₂, 12, 12) (dual of [(1398100, 12), 16777144, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(32⁵⁶, 8388600, F₃₂, 12) (dual of [8388600, 8388544, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁵⁶, large, F₃₂, 12) (dual of [large, large−56, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(59, 71, 1409025)-Net in Base 32 — Constructive

(59, 71, 1409025)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (11, 17, 10925)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32¹⁷, 10925, F₃₂, 6, 6) (dual of [(10925, 6), 65533, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(32¹⁷, 32775, F₃₂, 6) (dual of [32775, 32758, 7]-code), using
      - constructionÂ X applied to C_e(5) âŠ‚ C_e(3) [i] based on
        linear OA(32¹⁶, 32768, F₃₂, 6) (dual of [32768, 32752, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(32¹⁰, 32768, F₃₂, 4) (dual of [32768, 32758, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [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. (42, 54, 1398100)-net in base 32, using
  - base change [i] based on digital (33, 45, 1398100)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁴⁵, 1398100, F₆₄, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
      - OA 6-folding and stacking [i] based on linear OA(64⁴⁵, 8388600, F₆₄, 12) (dual of [8388600, 8388555, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴⁵, large, F₆₄, 12) (dual of [large, large−45, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(59, 71, large)-Net over F₃₂ — Digital

Digital (59, 71, large)-net over F₃₂, using

t-expansion [i] based on digital (56, 71, large)-net over F₃₂, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁷¹, large, F₃₂, 15) (dual of [large, large−71, 16]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(59, 71, large)-Net in Base 32 — Upper bound on s

There is no (59, 71, large)-net in base 32, because

10 times m-reduction [i] would yield (59, 61, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (59, 71, s)-Nets in Base 32

(59, 71, 1399167)-Net over F32 — Constructive and digital

(59, 71, 1409025)-Net in Base 32 — Constructive

(59, 71, large)-Net over F32 — Digital

(59, 71, large)-Net in Base 32 — Upper bound on s

(59, 71, 1399167)-Net over F₃₂ — Constructive and digital

(59, 71, large)-Net over F₃₂ — Digital