MinT - Best Known (58, 58+32, s)-Nets in Base 16

Best Known (58, 58+32, s)-Nets in Base 16

(58, 58+32, 583)-Net over F₁₆ — Constructive and digital

Digital (58, 90, 583)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 22, 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 (36, 68, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 34, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(58, 58+32, 594)-Net in Base 16 — Constructive

(58, 90, 594)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (10, 26, 80)-net in base 16, using
  - 1 times m-reduction [i] based on (10, 27, 80)-net in base 16, using
    - base change [i] based on digital (1, 18, 80)-net over F₆₄, using
      - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
2. digital (32, 64, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 32, 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]

(58, 58+32, 2983)-Net over F₁₆ — Digital

Digital (58, 90, 2983)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹⁰, 2983, F₁₆, 32) (dual of [2983, 2893, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁹⁰, 4096, F₁₆, 32) (dual of [4096, 4006, 33]-code), using
  - 1 times truncation [i] based on linear OA(16⁹¹, 4097, F₁₆, 33) (dual of [4097, 4006, 34]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 4097Â | 16⁶âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]

(58, 58+32, 2689186)-Net in Base 16 — Upper bound on s

There is no (58, 90, 2689187)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 348544 193943 068420 785726 707084 854264 773354 053286 180281 018757 888396 988853 706569 742493 852414 935991 887863 149006 > 16⁹⁰ [i]