MinT - Best Known (79, 79+43, s)-Nets in Base 16

Best Known (79, 79+43, s)-Nets in Base 16

(79, 79+43, 587)-Net over F₁₆ — Constructive and digital

Digital (79, 122, 587)-net over F₁₆, using

16¹ times duplication [i] based on 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]

(79, 79+43, 618)-Net in Base 16 — Constructive

(79, 122, 618)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (15, 36, 104)-net in base 16, using
  - base change [i] based on digital (3, 24, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, 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]

(79, 79+43, 3828)-Net over F₁₆ — Digital

Digital (79, 122, 3828)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹²², 3828, F₁₆, 43) (dual of [3828, 3706, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(16¹²², 4104, F₁₆, 43) (dual of [4104, 3982, 44]-code), using
  - constructionÂ X applied to C([0,21]) âŠ‚ C([0,20]) [i] based on
    1. linear OA(16¹²¹, 4097, F₁₆, 43) (dual of [4097, 3976, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 16⁶âˆ’1, defining interval IÂ =Â [0,21], and minimum distance dÂ â‰¥ |{−21,−20,…,21}|+1Â =Â 44 (BCH-bound) [i]
    2. linear OA(16¹¹⁵, 4097, F₁₆, 41) (dual of [4097, 3982, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 16⁶âˆ’1, defining interval IÂ =Â [0,20], and minimum distance dÂ â‰¥ |{−20,−19,…,20}|+1Â =Â 42 (BCH-bound) [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]

(79, 79+43, 5016779)-Net in Base 16 — Upper bound on s

There is no (79, 122, 5016780)-net in base 16, because

1 times m-reduction [i] would yield (79, 121, 5016780)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 49 948153 470565 925921 013035 263569 912237 827899 933689 804508 343341 568512 928891 852916 987159 165075 759564 377128 611678 671455 648567 549392 564021 463554 697576 > 16¹²¹ [i]