MinT - Best Known (218−40, 218, s)-Nets in Base 4

Best Known (218−40, 218, s)-Nets in Base 4

(218−40, 218, 1539)-Net over F₄ — Constructive and digital

Digital (178, 218, 1539)-net over F₄, using

7 times m-reduction [i] based on digital (178, 225, 1539)-net over F₄, using
- trace code for nets [i] based on digital (28, 75, 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]

(218−40, 218, 13702)-Net over F₄ — Digital

Digital (178, 218, 13702)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²¹⁸, 13702, F₄, 40) (dual of [13702, 13484, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(4²¹⁸, 16420, F₄, 40) (dual of [16420, 16202, 41]-code), using
  - constructionÂ X applied to C([0,20]) âŠ‚ C([0,17]) [i] based on
    1. 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]
    2. linear OA(4¹⁸³, 16385, F₄, 35) (dual of [16385, 16202, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,17], and minimum distance dÂ â‰¥ |{−17,−16,…,17}|+1Â =Â 36 (BCH-bound) [i]
    3. linear OA(4⁷, 35, F₄, 4) (dual of [35, 28, 5]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁷, 43, F₄, 4) (dual of [43, 36, 5]-code), using
        quaternary constacyclic codes with minimum distance 5 [i]

(218−40, 218, large)-Net in Base 4 — Upper bound on s

There is no (178, 218, large)-net in base 4, because

38 times m-reduction [i] would yield (178, 180, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]