MinT - Best Known (69, 83, s)-Nets in Base 25

Best Known (69, 83, s)-Nets in Base 25

(69, 83, 1199022)-Net over F₂₅ — Constructive and digital

Digital (69, 83, 1199022)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (10, 17, 651)-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, 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]
    20. digital (0, 1, 26)-net over F₂₅ (see above)
    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 (1, 8, 27)-net over F₂₅, using
      - net from sequence [i] based on digital (1, 26)-sequence over F₂₅, using
        Niederreiter sequence [i]
2. digital (52, 66, 1198371)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁶⁶, 1198371, F₂₅, 14, 14) (dual of [(1198371, 14), 16777128, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(25⁶⁶, 8388597, F₂₅, 14) (dual of [8388597, 8388531, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁶⁶, large, F₂₅, 14) (dual of [large, large−66, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(69, 83, large)-Net over F₂₅ — Digital

Digital (69, 83, large)-net over F₂₅, using

4 times m-reduction [i] based on digital (69, 87, large)-net over F₂₅, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁸⁷, large, F₂₅, 18) (dual of [large, large−87, 19]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(25⁸⁶, large, F₂₅, 18) (dual of [large, large−86, 19]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(69, 83, large)-Net in Base 25 — Upper bound on s

There is no (69, 83, large)-net in base 25, because

12 times m-reduction [i] would yield (69, 71, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (69, 83, s)-Nets in Base 25

(69, 83, 1199022)-Net over F25 — Constructive and digital

(69, 83, large)-Net over F25 — Digital

(69, 83, large)-Net in Base 25 — Upper bound on s

(69, 83, 1199022)-Net over F₂₅ — Constructive and digital

(69, 83, large)-Net over F₂₅ — Digital