MinT - Best Known (242−43, 242, s)-Nets in Base 4

Best Known (242−43, 242, s)-Nets in Base 4

(242−43, 242, 1560)-Net over F₄ — Constructive and digital

Digital (199, 242, 1560)-net over F₄, using

4¹ times duplication [i] based on digital (198, 241, 1560)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 28, 21)-net over F₄, using
    - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
        an explicitly constructive algebraic function field [i]
  2. digital (170, 213, 1539)-net over F₄, using
    - trace code for nets [i] based on digital (28, 71, 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]

(242−43, 242, 16447)-Net over F₄ — Digital

Digital (199, 242, 16447)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁴², 16447, F₄, 43) (dual of [16447, 16205, 44]-code), using
- constructionÂ X applied to C_e(42) âŠ‚ C_e(33) [i] based on
  1. linear OA(4²²⁵, 16384, F₄, 43) (dual of [16384, 16159, 44]-code), using an extension C_e(42) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,42], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 43 [i]
  2. linear OA(4¹⁷⁶, 16384, F₄, 34) (dual of [16384, 16208, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
  3. linear OA(4¹⁷, 63, F₄, 8) (dual of [63, 46, 9]-code), using
    - a code from Grasslâ€™s database [i]

(242−43, 242, large)-Net in Base 4 — Upper bound on s

There is no (199, 242, large)-net in base 4, because

41 times m-reduction [i] would yield (199, 201, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]