MinT - Best Known (82, 82+13, s)-Nets in Base 25

Best Known (82, 82+13, s)-Nets in Base 25

(82, 82+13, 2796850)-Net over F₂₅ — Constructive and digital

Digital (82, 95, 2796850)-net over F₂₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 8, 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, 0, 26)-net over F₂₅ (see above)
    21. digital (0, 0, 26)-net over F₂₅ (see above)
    22. 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]
    23. digital (0, 1, 26)-net over F₂₅ (see above)
    24. digital (0, 2, 26)-net over F₂₅, using
      - kÂ =Â 2 construction [i]
    25. digital (0, 4, 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]
2. digital (20, 26, 1398100)-net over F₂₅, using
  - s-reduction based on digital (20, 26, 2796201)-net over F₂₅, using
    - net defined by OOA [i] based on linear OOA(25²⁶, 2796201, F₂₅, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(25²⁶, large, F₂₅, 6) (dual of [large, large−26, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
3. digital (48, 61, 1398100)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁶¹, 1398100, F₂₅, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(25⁶¹, 8388601, F₂₅, 13) (dual of [8388601, 8388540, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁶¹, large, F₂₅, 13) (dual of [large, large−61, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(82, 82+13, large)-Net over F₂₅ — Digital

Digital (82, 95, large)-net over F₂₅, using

t-expansion [i] based on digital (81, 95, large)-net over F₂₅, using
- 7 times m-reduction [i] based on digital (81, 102, large)-net over F₂₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25¹⁰², large, F₂₅, 21) (dual of [large, large−102, 22]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(25¹⁰¹, large, F₂₅, 21) (dual of [large, large−101, 22]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(82, 82+13, large)-Net in Base 25 — Upper bound on s

There is no (82, 95, large)-net in base 25, because

11 times m-reduction [i] would yield (82, 84, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (82, 82+13, s)-Nets in Base 25

(82, 82+13, 2796850)-Net over F25 — Constructive and digital

(82, 82+13, large)-Net over F25 — Digital

(82, 82+13, large)-Net in Base 25 — Upper bound on s

(82, 82+13, 2796850)-Net over F₂₅ — Constructive and digital

(82, 82+13, large)-Net over F₂₅ — Digital