MinT - Best Known (61, 61+30, s)-Nets in Base 16

Best Known (61, 61+30, s)-Nets in Base 16

(61, 61+30, 1028)-Net over F₁₆ — Constructive and digital

Digital (61, 91, 1028)-net over F₁₆, using

1 times m-reduction [i] based on digital (61, 92, 1028)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. 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, 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 (31, 62, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 31, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(61, 61+30, 4703)-Net over F₁₆ — Digital

Digital (61, 91, 4703)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹¹, 4703, F₁₆, 30) (dual of [4703, 4612, 31]-code), using
- 598 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 14 times 0, 1, 57 times 0, 1, 175 times 0, 1, 344 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]

(61, 61+30, large)-Net in Base 16 — Upper bound on s

There is no (61, 91, large)-net in base 16, because

28 times m-reduction [i] would yield (61, 63, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]