MinT - Best Known (58, 58+33, s)-Nets in Base 16

Best Known (58, 58+33, s)-Nets in Base 16

(58, 58+33, 581)-Net over F₁₆ — Constructive and digital

Digital (58, 91, 581)-net over F₁₆, using

16¹ times duplication [i] based on digital (57, 90, 581)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 22, 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 (35, 68, 516)-net over F₁₆, using
    - trace code for nets [i] based on digital (1, 34, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(58, 58+33, 2577)-Net over F₁₆ — Digital

Digital (58, 91, 2577)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹¹, 2577, F₁₆, 33) (dual of [2577, 2486, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁹¹, 4096, F₁₆, 33) (dual of [4096, 4005, 34]-code), using
  - an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]

(58, 58+33, 2689186)-Net in Base 16 — Upper bound on s

There is no (58, 91, 2689187)-net in base 16, because

1 times m-reduction [i] would yield (58, 90, 2689187)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 348544 193943 068420 785726 707084 854264 773354 053286 180281 018757 888396 988853 706569 742493 852414 935991 887863 149006 > 16⁹⁰ [i]