MinT - Best Known (194, 194+42, s)-Nets in Base 4

Best Known (194, 194+42, s)-Nets in Base 4

(194, 194+42, 1556)-Net over F₄ — Constructive and digital

Digital (194, 236, 1556)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 26, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (168, 210, 1539)-net over F₄, using
  - trace code for nets [i] based on digital (28, 70, 513)-net over F₆₄, using
    - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(194, 194+42, 16451)-Net over F₄ — Digital

Digital (194, 236, 16451)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²³⁶, 16451, F₄, 42) (dual of [16451, 16215, 43]-code), using
- constructionÂ X applied to C_e(41) âŠ‚ C_e(32) [i] based on
  1. linear OA(4²¹⁸, 16384, F₄, 42) (dual of [16384, 16166, 43]-code), using an extension C_e(41) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,41], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 42 [i]
  2. linear OA(4¹⁶⁹, 16384, F₄, 33) (dual of [16384, 16215, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]
  3. linear OA(4¹⁸, 67, F₄, 8) (dual of [67, 49, 9]-code), using
    - discarding factors / shortening the dual code based on linear OA(4¹⁸, 68, F₄, 8) (dual of [68, 50, 9]-code), using
      - a code from Grasslâ€™s database [i]

(194, 194+42, large)-Net in Base 4 — Upper bound on s

There is no (194, 236, large)-net in base 4, because

40 times m-reduction [i] would yield (194, 196, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]