MinT - Best Known (81, 81+16, s)-Nets in Base 32

Best Known (81, 81+16, s)-Nets in Base 32

(81, 81+16, 1049642)-Net over F₃₂ — Constructive and digital

Digital (81, 97, 1049642)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (13, 21, 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, 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]
    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, 2, 33)-net over F₃₂ (see above)
    31. digital (0, 4, 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, 9, 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 (60, 76, 1048575)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁷⁶, 1048575, F₃₂, 16, 16) (dual of [(1048575, 16), 16777124, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(32⁷⁶, 8388600, F₃₂, 16) (dual of [8388600, 8388524, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁷⁶, large, F₃₂, 16) (dual of [large, large−76, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(81, 81+16, 1056768)-Net in Base 32 — Constructive

(81, 97, 1056768)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (15, 23, 8193)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32²³, 8193, F₃₂, 8, 8) (dual of [(8193, 8), 65521, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(32²³, 32772, F₃₂, 8) (dual of [32772, 32749, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(32²³, 32775, F₃₂, 8) (dual of [32775, 32752, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(5) [i] based on
        linear OA(32²², 32768, F₃₂, 8) (dual of [32768, 32746, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        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(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. (58, 74, 1048575)-net in base 32, using
  - net defined by OOA [i] based on OOA(32⁷⁴, 1048575, S₃₂, 16, 16), using
    - OA 8-folding and stacking [i] based on OA(32⁷⁴, 8388600, S₃₂, 16), using
      - discarding factors based on OA(32⁷⁴, large, S₃₂, 16), using
        discarding parts of the base [i] based on linear OA(64⁶¹, large, F₆₄, 16) (dual of [large, large−61, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(81, 81+16, large)-Net over F₃₂ — Digital

Digital (81, 97, large)-net over F₃₂, using

t-expansion [i] based on digital (79, 97, large)-net over F₃₂, using
- 3 times m-reduction [i] based on digital (79, 100, large)-net over F₃₂, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(81, 81+16, large)-Net in Base 32 — Upper bound on s

There is no (81, 97, large)-net in base 32, because

14 times m-reduction [i] would yield (81, 83, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (81, 81+16, s)-Nets in Base 32

(81, 81+16, 1049642)-Net over F32 — Constructive and digital

(81, 81+16, 1056768)-Net in Base 32 — Constructive

(81, 81+16, large)-Net over F32 — Digital

(81, 81+16, large)-Net in Base 32 — Upper bound on s

(81, 81+16, 1049642)-Net over F₃₂ — Constructive and digital

(81, 81+16, large)-Net over F₃₂ — Digital