MinT - Best Known (26−8, 26, s)-Nets in Base 16

Best Known (26−8, 26, s)-Nets in Base 16

(26−8, 26, 1045)-Net over F₁₆ — Constructive and digital

Digital (18, 26, 1045)-net over F₁₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 2, 17)-net over F₁₆, using
  - kÂ =Â 2 construction [i]
2. digital (4, 8, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 4, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]
3. digital (8, 16, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 8, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(26−8, 26, 1056)-Net in Base 16 — Constructive

(18, 26, 1056)-net in base 16, using

16¹ times duplication [i] based on (17, 25, 1056)-net in base 16, using
- base change [i] based on digital (12, 20, 1056)-net over F₃₂, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 33)-net over F₃₂, using
      - s-reduction based on digital (0, 0, s)-net over F₃₂ with arbitrarily large s, using
        arbitrary set of 32⁰ points [i]
    2. digital (0, 0, 33)-net over F₃₂ (see above)
    3. digital (0, 0, 33)-net over F₃₂ (see above)
    4. digital (0, 0, 33)-net over F₃₂ (see above)
    5. digital (0, 0, 33)-net over F₃₂ (see above)
    6. digital (0, 0, 33)-net over F₃₂ (see above)
    7. digital (0, 0, 33)-net over F₃₂ (see above)
    8. digital (0, 0, 33)-net over F₃₂ (see above)
    9. digital (0, 0, 33)-net over F₃₂ (see above)
    10. digital (0, 0, 33)-net over F₃₂ (see above)
    11. digital (0, 0, 33)-net over F₃₂ (see above)
    12. digital (0, 0, 33)-net over F₃₂ (see above)
    13. digital (0, 0, 33)-net over F₃₂ (see above)
    14. digital (0, 0, 33)-net over F₃₂ (see above)
    15. digital (0, 0, 33)-net over F₃₂ (see above)
    16. digital (0, 0, 33)-net over F₃₂ (see above)
    17. digital (0, 0, 33)-net over F₃₂ (see above)
    18. digital (0, 0, 33)-net over F₃₂ (see above)
    19. digital (0, 0, 33)-net over F₃₂ (see above)
    20. digital (0, 0, 33)-net over F₃₂ (see above)
    21. digital (0, 0, 33)-net over F₃₂ (see above)
    22. digital (0, 0, 33)-net over F₃₂ (see above)
    23. digital (0, 0, 33)-net over F₃₂ (see above)
    24. digital (0, 0, 33)-net over F₃₂ (see above)
    25. digital (0, 1, 33)-net over F₃₂, using
      - s-reduction based on digital (0, 1, s)-net over F₃₂ with arbitrarily large s, using
        construction for strength kÂ =Â 1 [i]
    26. digital (0, 1, 33)-net over F₃₂ (see above)
    27. digital (0, 1, 33)-net over F₃₂ (see above)
    28. digital (0, 1, 33)-net over F₃₂ (see above)
    29. digital (0, 2, 33)-net over F₃₂, using
      - kÂ =Â 2 construction [i]
    30. digital (0, 2, 33)-net over F₃₂ (see above)
    31. digital (0, 4, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
        the rational function field F₃₂(x) [i]
        Niederreiter sequence [i]
    32. digital (0, 8, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)

(26−8, 26, 6722)-Net over F₁₆ — Digital

Digital (18, 26, 6722)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16²⁶, 6722, F₁₆, 8) (dual of [6722, 6696, 9]-code), using
- 2619 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 0, 0, 0, 1, 61 times 0, 1, 611 times 0, 1, 1940 times 0) [i] based on linear OA(16²², 4099, F₁₆, 8) (dual of [4099, 4077, 9]-code), using
  - constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
    1. linear OA(16²², 4096, F₁₆, 8) (dual of [4096, 4074, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
    2. linear OA(16¹⁹, 4096, F₁₆, 7) (dual of [4096, 4077, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
    3. linear OA(16⁰, 3, F₁₆, 0) (dual of [3, 3, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(16⁰, s, F₁₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(26−8, 26, large)-Net in Base 16 — Upper bound on s

There is no (18, 26, large)-net in base 16, because

6 times m-reduction [i] would yield (18, 20, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (26−8, 26, s)-Nets in Base 16

(26−8, 26, 1045)-Net over F16 — Constructive and digital

(26−8, 26, 1056)-Net in Base 16 — Constructive

(26−8, 26, 6722)-Net over F16 — Digital

(26−8, 26, large)-Net in Base 16 — Upper bound on s

(26−8, 26, 1045)-Net over F₁₆ — Constructive and digital

(26−8, 26, 6722)-Net over F₁₆ — Digital