MinT - Best Known (70−12, 70, s)-Nets in Base 25

Best Known (70−12, 70, s)-Nets in Base 25

(70−12, 70, 1398750)-Net over F₂₅ — Constructive and digital

Digital (58, 70, 1398750)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (8, 14, 650)-net over F₂₅, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 26)-net over F₂₅, using
      - s-reduction based on digital (0, 0, s)-net over F₂₅ with arbitrarily large s, using
        arbitrary set of 25⁰ points [i]
    2. digital (0, 0, 26)-net over F₂₅ (see above)
    3. digital (0, 0, 26)-net over F₂₅ (see above)
    4. digital (0, 0, 26)-net over F₂₅ (see above)
    5. digital (0, 0, 26)-net over F₂₅ (see above)
    6. digital (0, 0, 26)-net over F₂₅ (see above)
    7. digital (0, 0, 26)-net over F₂₅ (see above)
    8. digital (0, 0, 26)-net over F₂₅ (see above)
    9. digital (0, 0, 26)-net over F₂₅ (see above)
    10. digital (0, 0, 26)-net over F₂₅ (see above)
    11. digital (0, 0, 26)-net over F₂₅ (see above)
    12. digital (0, 0, 26)-net over F₂₅ (see above)
    13. digital (0, 0, 26)-net over F₂₅ (see above)
    14. digital (0, 0, 26)-net over F₂₅ (see above)
    15. digital (0, 0, 26)-net over F₂₅ (see above)
    16. digital (0, 0, 26)-net over F₂₅ (see above)
    17. digital (0, 0, 26)-net over F₂₅ (see above)
    18. digital (0, 0, 26)-net over F₂₅ (see above)
    19. digital (0, 0, 26)-net over F₂₅ (see above)
    20. digital (0, 1, 26)-net over F₂₅, using
      - s-reduction based on digital (0, 1, s)-net over F₂₅ with arbitrarily large s, using
        construction for strength kÂ =Â 1 [i]
    21. digital (0, 1, 26)-net over F₂₅ (see above)
    22. digital (0, 1, 26)-net over F₂₅ (see above)
    23. digital (0, 2, 26)-net over F₂₅, using
      - kÂ =Â 2 construction [i]
    24. digital (0, 3, 26)-net over F₂₅, using
      - net from sequence [i] based on digital (0, 25)-sequence over F₂₅, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 0 and N(F) ≥ 26, using
        the rational function field F₂₅(x) [i]
        Niederreiter sequence [i]
    25. digital (0, 6, 26)-net over F₂₅, using
      - net from sequence [i] based on digital (0, 25)-sequence over F₂₅ (see above)
2. digital (44, 56, 1398100)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁵⁶, 1398100, F₂₅, 12, 12) (dual of [(1398100, 12), 16777144, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(25⁵⁶, 8388600, F₂₅, 12) (dual of [8388600, 8388544, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁵⁶, large, F₂₅, 12) (dual of [large, large−56, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(70−12, 70, large)-Net over F₂₅ — Digital

Digital (58, 70, large)-net over F₂₅, using

t-expansion [i] based on digital (57, 70, large)-net over F₂₅, using
- 2 times m-reduction [i] based on digital (57, 72, large)-net over F₂₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁷², large, F₂₅, 15) (dual of [large, large−72, 16]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(25⁷¹, large, F₂₅, 15) (dual of [large, large−71, 16]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(70−12, 70, large)-Net in Base 25 — Upper bound on s

There is no (58, 70, large)-net in base 25, because

10 times m-reduction [i] would yield (58, 60, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (70−12, 70, s)-Nets in Base 25

(70−12, 70, 1398750)-Net over F25 — Constructive and digital

(70−12, 70, large)-Net over F25 — Digital

(70−12, 70, large)-Net in Base 25 — Upper bound on s

(70−12, 70, 1398750)-Net over F₂₅ — Constructive and digital

(70−12, 70, large)-Net over F₂₅ — Digital