MinT - Best Known (77, 77+40, s)-Nets in Base 16

Best Known (77, 77+40, s)-Nets in Base 16

(77, 77+40, 603)-Net over F₁₆ — Constructive and digital

Digital (77, 117, 603)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (17, 37, 89)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 11, 24)-net over F₁₆, using
      - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
        a function field by SÃ©mirat [i]
    2. digital (6, 26, 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 (40, 80, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 40, 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]

(77, 77+40, 645)-Net in Base 16 — Constructive

(77, 117, 645)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (15, 35, 129)-net in base 16, using
  - base change [i] based on digital (0, 20, 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 (42, 82, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 41, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(77, 77+40, 4293)-Net over F₁₆ — Digital

Digital (77, 117, 4293)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹¹⁷, 4293, F₁₆, 40) (dual of [4293, 4176, 41]-code), using
- 189 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 1, 10 times 0, 1, 43 times 0, 1, 130 times 0) [i] based on linear OA(16¹¹², 4099, F₁₆, 40) (dual of [4099, 3987, 41]-code), using
  - constructionÂ X applied to C_e(39) âŠ‚ C_e(38) [i] based on
    1. linear OA(16¹¹², 4096, F₁₆, 40) (dual of [4096, 3984, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
    2. linear OA(16¹⁰⁹, 4096, F₁₆, 39) (dual of [4096, 3987, 40]-code), using an extension C_e(38) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [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]

(77, 77+40, 6127962)-Net in Base 16 — Upper bound on s

There is no (77, 117, 6127963)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 762 146760 038137 720575 233028 052519 534960 907153 168854 061669 431310 760711 095210 571539 901998 133336 219065 533377 307248 384851 618134 092365 892732 288526 > 16¹¹⁷ [i]