MinT - Best Known (79, 79+12, s)-Nets in Base 32

Best Known (79, 79+12, s)-Nets in Base 32

(79, 79+12, 2797267)-Net over F₃₂ — Constructive and digital

Digital (79, 91, 2797267)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (5, 9, 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, 0, 33)-net over F₃₂ (see above)
    28. digital (0, 0, 33)-net over F₃₂ (see above)
    29. 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]
    30. digital (0, 1, 33)-net over F₃₂ (see above)
    31. digital (0, 2, 33)-net over F₃₂, using
      - kÂ =Â 2 construction [i]
    32. digital (1, 5, 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 (20, 26, 1398100)-net over F₃₂, using
  - s-reduction based on digital (20, 26, 2796201)-net over F₃₂, using
    - net defined by OOA [i] based on linear OOA(32²⁶, 2796201, F₃₂, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(32²⁶, large, F₃₂, 6) (dual of [large, large−26, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
3. 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]

(79, 79+12, 2828840)-Net in Base 32 — Constructive

(79, 91, 2828840)-net in base 32, using

base change [i] based on (53, 65, 2828840)-net in base 128, using
- 128¹ times duplication [i] based on (52, 64, 2828840)-net in base 128, using
  - base change [i] based on digital (44, 56, 2828840)-net over F₂₅₆, using
    - generalized (u,Â u+v)-construction [i] based on
      1. digital (2, 6, 32640)-net over F₂₅₆, using
        net defined by OOA [i] based on linear OOA(256⁶, 32640, F₂₅₆, 4, 4) (dual of [(32640, 4), 130554, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(256⁶, 65280, F₂₅₆, 4) (dual of [65280, 65274, 5]-code), using
        1 times truncation [i] based on linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 6]-code), using
        a code by Danev and Olsson [i]
      2. digital (10, 16, 1398100)-net over F₂₅₆, using
        s-reduction based on digital (10, 16, 2796201)-net over F₂₅₆, using
        net defined by OOA [i] based on linear OOA(256¹⁶, 2796201, F₂₅₆, 6, 6) (dual of [(2796201, 6), 16777190, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(256¹⁶, large, F₂₅₆, 6) (dual of [large, large−16, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
      3. digital (22, 34, 1398100)-net over F₂₅₆, using
        net defined by OOA [i] based on linear OOA(256³⁴, 1398100, F₂₅₆, 12, 12) (dual of [(1398100, 12), 16777166, 13]-NRT-code), using
        OA 6-folding and stacking [i] based on linear OA(256³⁴, 8388600, F₂₅₆, 12) (dual of [8388600, 8388566, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(256³⁴, large, F₂₅₆, 12) (dual of [large, large−34, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(79, 79+12, large)-Net over F₃₂ — Digital

Digital (79, 91, large)-net over F₃₂, using

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

(79, 79+12, large)-Net in Base 32 — Upper bound on s

There is no (79, 91, large)-net in base 32, because

10 times m-reduction [i] would yield (79, 81, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (79, 79+12, s)-Nets in Base 32

(79, 79+12, 2797267)-Net over F32 — Constructive and digital

(79, 79+12, 2828840)-Net in Base 32 — Constructive

(79, 79+12, large)-Net over F32 — Digital

(79, 79+12, large)-Net in Base 32 — Upper bound on s

(79, 79+12, 2797267)-Net over F₃₂ — Constructive and digital

(79, 79+12, large)-Net over F₃₂ — Digital