MinT - Best Known (85, 85+17, s)-Nets in Base 27

Best Known (85, 85+17, s)-Nets in Base 27

(85, 85+17, 1049341)-Net over F₂₇ — Constructive and digital

Digital (85, 102, 1049341)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (13, 21, 766)-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 with 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, 1, 28)-net over F₂₇, using
      - s-reduction based on digital (0, 1, s)-net over F₂₇ with arbitrarily large s, using
        construction for strength kÂ =Â 1 [i]
    21. digital (0, 1, 28)-net over F₂₇ (see above)
    22. digital (0, 1, 28)-net over F₂₇ (see above)
    23. digital (0, 1, 28)-net over F₂₇ (see above)
    24. digital (0, 2, 28)-net over F₂₇, using
      - kÂ =Â 2 construction [i]
    25. digital (0, 2, 28)-net over F₂₇ (see above)
    26. 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]
    27. digital (1, 9, 38)-net over F₂₇, using
      - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]
2. digital (64, 81, 1048575)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁸¹, 1048575, F₂₇, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(27⁸¹, 8388601, F₂₇, 17) (dual of [8388601, 8388520, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁸¹, large, F₂₇, 17) (dual of [large, large−81, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(85, 85+17, 1050216)-Net in Base 27 — Constructive

(85, 102, 1050216)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. (13, 21, 1641)-net in base 27, using
  - net defined by OOA [i] based on OOA(27²¹, 1641, S₂₇, 8, 8), using
    - OA 4-folding and stacking [i] based on OA(27²¹, 6564, S₂₇, 8), using
      - 1 times code embedding in larger space [i] based on OA(27²⁰, 6563, S₂₇, 8), using
        discarding parts of the base [i] based on linear OA(81¹⁵, 6563, F₈₁, 8) (dual of [6563, 6548, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        linear OA(81¹⁵, 6561, F₈₁, 8) (dual of [6561, 6546, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(81¹³, 6561, F₈₁, 7) (dual of [6561, 6548, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(81⁰, 2, F₈₁, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁰, s, F₈₁, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. digital (64, 81, 1048575)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁸¹, 1048575, F₂₇, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(27⁸¹, 8388601, F₂₇, 17) (dual of [8388601, 8388520, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁸¹, large, F₂₇, 17) (dual of [large, large−81, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(85, 85+17, large)-Net over F₂₇ — Digital

Digital (85, 102, large)-net over F₂₇, using

t-expansion [i] based on digital (84, 102, large)-net over F₂₇, using
- 4 times m-reduction [i] based on digital (84, 106, large)-net over F₂₇, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27¹⁰⁶, large, F₂₇, 22) (dual of [large, large−106, 23]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(85, 85+17, large)-Net in Base 27 — Upper bound on s

There is no (85, 102, large)-net in base 27, because

15 times m-reduction [i] would yield (85, 87, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (85, 85+17, s)-Nets in Base 27

(85, 85+17, 1049341)-Net over F27 — Constructive and digital

(85, 85+17, 1050216)-Net in Base 27 — Constructive

(85, 85+17, large)-Net over F27 — Digital

(85, 85+17, large)-Net in Base 27 — Upper bound on s

(85, 85+17, 1049341)-Net over F₂₇ — Constructive and digital

(85, 85+17, large)-Net over F₂₇ — Digital