MinT - Best Known (78, 121, s)-Nets in Base 16

Best Known (78, 121, s)-Nets in Base 16

(78, 121, 587)-Net over F₁₆ — Constructive and digital

Digital (78, 121, 587)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 27, 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 (51, 94, 522)-net over F₁₆, using
  - trace code for nets [i] based on digital (4, 47, 261)-net over F₂₅₆, using
    - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(78, 121, 612)-Net in Base 16 — Constructive

(78, 121, 612)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (14, 35, 98)-net in base 16, using
  - base change [i] based on digital (7, 28, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]
2. digital (43, 86, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 43, 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]

(78, 121, 3576)-Net over F₁₆ — Digital

Digital (78, 121, 3576)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹²¹, 3576, F₁₆, 43) (dual of [3576, 3455, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(16¹²¹, 4096, F₁₆, 43) (dual of [4096, 3975, 44]-code), using
  - an extension C_e(42) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,42], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 43 [i]

(78, 121, 4396284)-Net in Base 16 — Upper bound on s

There is no (78, 121, 4396285)-net in base 16, because

1 times m-reduction [i] would yield (78, 120, 4396285)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 3 121754 844744 487220 916486 884833 238338 801637 016506 674469 451865 721870 726925 309907 452695 772650 911929 971926 834671 293609 610086 869668 589249 097035 634776 > 16¹²⁰ [i]