MinT - Best Known (79, 79+42, s)-Nets in Base 16

Best Known (79, 79+42, s)-Nets in Base 16

(79, 79+42, 596)-Net over F₁₆ — Constructive and digital

Digital (79, 121, 596)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (16, 37, 82)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 10, 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, 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 (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]

(79, 79+42, 618)-Net in Base 16 — Constructive

(79, 121, 618)-net in base 16, using

1 times m-reduction [i] based on (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+42, 4138)-Net over F₁₆ — Digital

Digital (79, 121, 4138)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹²¹, 4138, F₁₆, 42) (dual of [4138, 4017, 43]-code), using
- 36 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 6 times 0, 1, 28 times 0) [i] based on linear OA(16¹¹⁸, 4099, F₁₆, 42) (dual of [4099, 3981, 43]-code), using
  - constructionÂ X applied to C_e(41) âŠ‚ C_e(40) [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(16¹¹⁵, 4096, F₁₆, 41) (dual of [4096, 3981, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
    3. linear OA(16⁰, 3, F₁₆, 0) (dual of [3, 3, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(16⁰, s, F₁₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(79, 79+42, 5016779)-Net in Base 16 — Upper bound on s

There is no (79, 121, 5016780)-net in base 16, because

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]