MinT - Best Known (119−42, 119, s)-Nets in Base 16

Best Known (119−42, 119, s)-Nets in Base 16

(119−42, 119, 587)-Net over F₁₆ — Constructive and digital

Digital (77, 119, 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 (50, 92, 522)-net over F₁₆, using
  - trace code for nets [i] based on digital (4, 46, 261)-net over F₂₅₆, using
    - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(119−42, 119, 612)-Net in Base 16 — Constructive

(77, 119, 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 (42, 84, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 42, 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]

(119−42, 119, 3727)-Net over F₁₆ — Digital

Digital (77, 119, 3727)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹¹⁹, 3727, F₁₆, 42) (dual of [3727, 3608, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(16¹¹⁹, 4103, F₁₆, 42) (dual of [4103, 3984, 43]-code), using
  - constructionÂ X applied to C_e(41) âŠ‚ C_e(39) [i] based on
    1. linear OA(16¹¹⁸, 4096, F₁₆, 42) (dual of [4096, 3978, 43]-code), using an extension C_e(41) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,41], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 42 [i]
    2. 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]
    3. linear OA(16¹, 7, F₁₆, 1) (dual of [7, 6, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(16¹, s, F₁₆, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(119−42, 119, 3852534)-Net in Base 16 — Upper bound on s

There is no (77, 119, 3852535)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 195109 298626 196282 490932 470545 651600 923008 177534 779177 275546 186877 412378 950354 579063 700879 419211 575199 435133 393557 963297 975900 185500 978904 978526 > 16¹¹⁹ [i]