MinT - Best Known (58, 58+30, s)-Nets in Base 16

Best Known (58, 58+30, s)-Nets in Base 16

(58, 58+30, 596)-Net over F₁₆ — Constructive and digital

Digital (58, 88, 596)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (13, 28, 82)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 7, 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, 21, 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 (30, 60, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 30, 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+30, 664)-Net in Base 16 — Constructive

(58, 88, 664)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (13, 28, 150)-net in base 16, using
  - base change [i] based on digital (1, 16, 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 (30, 60, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 30, 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+30, 4121)-Net over F₁₆ — Digital

Digital (58, 88, 4121)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸⁸, 4121, F₁₆, 30) (dual of [4121, 4033, 31]-code), using
- 19 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 14 times 0) [i] based on linear OA(16⁸⁵, 4099, F₁₆, 30) (dual of [4099, 4014, 31]-code), using
  - constructionÂ X applied to C_e(29) âŠ‚ C_e(28) [i] based on
    1. linear OA(16⁸⁵, 4096, F₁₆, 30) (dual of [4096, 4011, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
    2. linear OA(16⁸², 4096, F₁₆, 29) (dual of [4096, 4014, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [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]

(58, 58+30, 4964154)-Net in Base 16 — Upper bound on s

There is no (58, 88, 4964155)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 9174 021925 691976 011992 215954 351713 813826 421755 171073 662418 970987 049407 495221 395769 623059 169707 517980 367376 > 16⁸⁸ [i]