MinT - Best Known (242−44, 242, s)-Nets in Base 4

Best Known (242−44, 242, s)-Nets in Base 4

(242−44, 242, 1554)-Net over F₄ — Constructive and digital

Digital (198, 242, 1554)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 26, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
      - an explicitly constructive algebraic function field [i]
2. digital (172, 216, 1539)-net over F₄, using
  - trace code for nets [i] based on digital (28, 72, 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]

(242−44, 242, 15647)-Net over F₄ — Digital

Digital (198, 242, 15647)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁴², 15647, F₄, 44) (dual of [15647, 15405, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁴², 16406, F₄, 44) (dual of [16406, 16164, 45]-code), using
  - constructionÂ X applied to C([0,22]) âŠ‚ C([0,20]) [i] based on
    1. linear OA(4²³⁹, 16385, F₄, 45) (dual of [16385, 16146, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,22], and minimum distance dÂ â‰¥ |{−22,−21,…,22}|+1Â =Â 46 (BCH-bound) [i]
    2. linear OA(4²¹¹, 16385, F₄, 41) (dual of [16385, 16174, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,20], and minimum distance dÂ â‰¥ |{−20,−19,…,20}|+1Â =Â 42 (BCH-bound) [i]
    3. linear OA(4³, 21, F₄, 2) (dual of [21, 18, 3]-code), using
      - Hamming code H(3,4) [i]

(242−44, 242, large)-Net in Base 4 — Upper bound on s

There is no (198, 242, large)-net in base 4, because

42 times m-reduction [i] would yield (198, 200, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]