MinT - Best Known (30, 45, s)-Nets in Base 16

Best Known (30, 45, s)-Nets in Base 16

(30, 45, 1028)-Net over F₁₆ — Constructive and digital

Digital (30, 45, 1028)-net over F₁₆, using

16¹ times duplication [i] based on digital (29, 44, 1028)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 14, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 7, 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]
  2. digital (15, 30, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 15, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(30, 45, 4115)-Net over F₁₆ — Digital

Digital (30, 45, 4115)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁴⁵, 4115, F₁₆, 15) (dual of [4115, 4070, 16]-code), using
- 14 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 13 times 0) [i] based on linear OA(16⁴³, 4099, F₁₆, 15) (dual of [4099, 4056, 16]-code), using
  - constructionÂ X applied to C_e(14) âŠ‚ C_e(13) [i] based on
    1. linear OA(16⁴³, 4096, F₁₆, 15) (dual of [4096, 4053, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
    2. linear OA(16⁴⁰, 4096, F₁₆, 14) (dual of [4096, 4056, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [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]

(30, 45, 8347979)-Net in Base 16 — Upper bound on s

There is no (30, 45, 8347980)-net in base 16, because

1 times m-reduction [i] would yield (30, 44, 8347980)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 95780 993787 759048 575643 824090 188665 437076 356685 807526 > 16⁴⁴ [i]