MinT - Information on Result #1231923

Information on Result #1231923

Digital (88, 101, 2797714)-net over F₂₇, using generalized (u,Â u+v)-construction based on

digital (0, 2, 28)-net over F₂₇, using
- kÂ =Â 2 construction [i]
digital (1, 4, 730)-net over F₂₇, using
- net defined by OOA [i] based on linear OOA(27⁴, 730, F₂₇, 3, 3) (dual of [(730, 3), 2186, 4]-NRT-code), using
  - appending kth column [i] based on linear OOA(27⁴, 730, F₂₇, 2, 3) (dual of [(730, 2), 1456, 4]-NRT-code), using
    - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(27⁴, 730, F₂₇, 3) (dual of [730, 726, 4]-code or 730-cap in PG(3,27)), using
      - ovoid in PG(3,Â 27) [i]
digital (4, 8, 756)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 0, 28)-net over F₂₇, using
    - s-reduction based on digital (0, 0, s)-net over F₂₇ for arbitrarily large s, using
      - arbitrary set of 27⁰ points [i]
  2. digital (0, 0, 28)-net over F₂₇ (see above)
  3. digital (0, 0, 28)-net over F₂₇ (see above)
  4. digital (0, 0, 28)-net over F₂₇ (see above)
  5. digital (0, 0, 28)-net over F₂₇ (see above)
  6. digital (0, 0, 28)-net over F₂₇ (see above)
  7. digital (0, 0, 28)-net over F₂₇ (see above)
  8. digital (0, 0, 28)-net over F₂₇ (see above)
  9. digital (0, 0, 28)-net over F₂₇ (see above)
  10. digital (0, 0, 28)-net over F₂₇ (see above)
  11. digital (0, 0, 28)-net over F₂₇ (see above)
  12. digital (0, 0, 28)-net over F₂₇ (see above)
  13. digital (0, 0, 28)-net over F₂₇ (see above)
  14. digital (0, 0, 28)-net over F₂₇ (see above)
  15. digital (0, 0, 28)-net over F₂₇ (see above)
  16. digital (0, 0, 28)-net over F₂₇ (see above)
  17. digital (0, 0, 28)-net over F₂₇ (see above)
  18. digital (0, 0, 28)-net over F₂₇ (see above)
  19. digital (0, 0, 28)-net over F₂₇ (see above)
  20. digital (0, 0, 28)-net over F₂₇ (see above)
  21. digital (0, 0, 28)-net over F₂₇ (see above)
  22. digital (0, 0, 28)-net over F₂₇ (see above)
  23. digital (0, 0, 28)-net over F₂₇ (see above)
  24. digital (0, 1, 28)-net over F₂₇, using
    - s-reduction based on digital (0, 1, s)-net over F₂₇ for arbitrarily large s, using
      - construction for strength kÂ =Â 1 [i]
  25. digital (0, 1, 28)-net over F₂₇ (see above)
  26. digital (0, 2, 28)-net over F₂₇ (see above)
  27. digital (0, 4, 28)-net over F₂₇, using
    - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
        the rational function field F₂₇(x) [i]
      - Niederreiter sequence [i]
digital (20, 26, 1398100)-net over F₂₇, using
- s-reduction based on digital (20, 26, 2796201)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27²⁶, 2796201, F₂₇, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(27²⁶, large, F₂₇, 6) (dual of [large, large−26, 7]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
digital (48, 61, 1398100)-net over F₂₇, using
- net defined by OOA [i] based on linear OOA(27⁶¹, 1398100, F₂₇, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
  - OOA 6-folding and stacking with additional row [i] based on linear OA(27⁶¹, 8388601, F₂₇, 13) (dual of [8388601, 8388540, 14]-code), using
    - discarding factors / shortening the dual code based on linear OA(27⁶¹, large, F₂₇, 13) (dual of [large, large−61, 14]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

Mode: Constructive and linear.

Optimality

Show details for fixed k and m, k and s, k and t, m and s, m and t, t and s.

Other Results with Identical Parameters

None.

Depending Results

None.