MinT - Best Known (94−32, 94, s)-Nets in Base 16

Best Known (94−32, 94, s)-Nets in Base 16

(94−32, 94, 596)-Net over F₁₆ — Constructive and digital

Digital (62, 94, 596)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (14, 30, 82)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 8, 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, 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 (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]

(94−32, 94, 664)-Net in Base 16 — Constructive

(62, 94, 664)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (14, 30, 150)-net in base 16, using
  - base change [i] based on (4, 20, 150)-net in base 64, using
    - 1 times m-reduction [i] based on (4, 21, 150)-net in base 64, using
      - base change [i] based on digital (1, 18, 150)-net over F₁₂₈, using
        net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, 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]

(94−32, 94, 4138)-Net over F₁₆ — Digital

Digital (62, 94, 4138)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹⁴, 4138, F₁₆, 32) (dual of [4138, 4044, 33]-code), using
- 38 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 6 times 0, 1, 29 times 0) [i] 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]

(94−32, 94, 5378382)-Net in Base 16 — Upper bound on s

There is no (62, 94, 5378383)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 153914 390711 462581 699662 033282 009576 415684 696425 504862 653704 456701 370481 416526 586604 780819 047490 470320 278363 655796 > 16⁹⁴ [i]