MinT - Best Known (84, 84+45, s)-Nets in Base 16

Best Known (84, 84+45, s)-Nets in Base 16

(84, 84+45, 596)-Net over F₁₆ — Constructive and digital

Digital (84, 129, 596)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (17, 39, 82)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 11, 17)-net over F₁₆, using
      - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
        the rational function field F₁₆(x) [i]
        Niederreiter sequence [i]
    2. digital (6, 28, 65)-net over F₁₆, using
      - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]
2. digital (45, 90, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 45, 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]

(84, 84+45, 643)-Net in Base 16 — Constructive

(84, 129, 643)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (17, 39, 129)-net in base 16, using
  - base change [i] based on (4, 26, 129)-net in base 64, using
    - 2 times m-reduction [i] based on (4, 28, 129)-net in base 64, using
      - base change [i] based on digital (0, 24, 129)-net over F₁₂₈, using
        net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]
2. digital (45, 90, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 45, 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]

(84, 84+45, 4115)-Net over F₁₆ — Digital

Digital (84, 129, 4115)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹²⁹, 4115, F₁₆, 45) (dual of [4115, 3986, 46]-code), using
- 14 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 13 times 0) [i] based on linear OA(16¹²⁷, 4099, F₁₆, 45) (dual of [4099, 3972, 46]-code), using
  - constructionÂ X applied to C_e(44) âŠ‚ C_e(43) [i] based on
    1. linear OA(16¹²⁷, 4096, F₁₆, 45) (dual of [4096, 3969, 46]-code), using an extension C_e(44) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,44], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 45 [i]
    2. linear OA(16¹²⁴, 4096, F₁₆, 44) (dual of [4096, 3972, 45]-code), using an extension C_e(43) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,43], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 44 [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]

(84, 84+45, 6117160)-Net in Base 16 — Upper bound on s

There is no (84, 129, 6117161)-net in base 16, because

1 times m-reduction [i] would yield (84, 128, 6117161)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 13407 809785 435751 774624 890224 279290 236230 701792 805513 458372 924061 691978 025674 192791 021669 704818 147371 138361 177978 329832 917563 423556 479084 609834 448962 922856 > 16¹²⁸ [i]