MinT - Best Known (68, 68+14, s)-Nets in Base 32

Best Known (68, 68+14, s)-Nets in Base 32

(68, 68+14, 1199427)-Net over F₃₂ — Constructive and digital

Digital (68, 82, 1199427)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (9, 16, 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, 0, 33)-net over F₃₂ (see above)
    25. digital (0, 0, 33)-net over F₃₂ (see above)
    26. 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]
    27. digital (0, 1, 33)-net over F₃₂ (see above)
    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 (0, 7, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
2. digital (52, 66, 1198371)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁶⁶, 1198371, F₃₂, 14, 14) (dual of [(1198371, 14), 16777128, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(32⁶⁶, 8388597, F₃₂, 14) (dual of [8388597, 8388531, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁶⁶, large, F₃₂, 14) (dual of [large, large−66, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(68, 68+14, 1199738)-Net in Base 32 — Constructive

(68, 82, 1199738)-net in base 32, using

32¹ times duplication [i] based on (67, 81, 1199738)-net in base 32, using
- (u,Â u+v)-construction [i] based on
  1. (10, 17, 1367)-net in base 32, using
    - net defined by OOA [i] based on OOA(32¹⁷, 1367, S₃₂, 7, 7), using
      - OOA 3-folding and stacking with additional row [i] based on OA(32¹⁷, 4102, S₃₂, 7), using
        discarding parts of the base [i] based on linear OA(64¹⁴, 4102, F₆₄, 7) (dual of [4102, 4088, 8]-code), using
        constructionÂ X applied to C([0,3]) âŠ‚ C([0,2]) [i] based on
        linear OA(64¹³, 4097, F₆₄, 7) (dual of [4097, 4084, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
        linear OA(64⁹, 4097, F₆₄, 5) (dual of [4097, 4088, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
        linear OA(64¹, 5, F₆₄, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹, s, F₆₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
  2. (50, 64, 1198371)-net in base 32, using
    - base change [i] based on digital (26, 40, 1198371)-net over F₂₅₆, using
      - net defined by OOA [i] based on linear OOA(256⁴⁰, 1198371, F₂₅₆, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
        OA 7-folding and stacking [i] based on linear OA(256⁴⁰, 8388597, F₂₅₆, 14) (dual of [8388597, 8388557, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁰, large, F₂₅₆, 14) (dual of [large, large−40, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(68, 68+14, large)-Net over F₃₂ — Digital

Digital (68, 82, large)-net over F₃₂, using

4 times m-reduction [i] based on digital (68, 86, large)-net over F₃₂, using
- Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+14, large)-Net in Base 32 — Upper bound on s

There is no (68, 82, large)-net in base 32, because

12 times m-reduction [i] would yield (68, 70, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (68, 68+14, s)-Nets in Base 32

(68, 68+14, 1199427)-Net over F32 — Constructive and digital

(68, 68+14, 1199738)-Net in Base 32 — Constructive

(68, 68+14, large)-Net over F32 — Digital

(68, 68+14, large)-Net in Base 32 — Upper bound on s

(68, 68+14, 1199427)-Net over F₃₂ — Constructive and digital

(68, 68+14, large)-Net over F₃₂ — Digital