MinT - Best Known (198−37, 198, s)-Nets in Base 4

Best Known (198−37, 198, s)-Nets in Base 4

(198−37, 198, 1539)-Net over F₄ — Constructive and digital

Digital (161, 198, 1539)-net over F₄, using

t-expansion [i] based on digital (160, 198, 1539)-net over F₄, using
- trace code for nets [i] based on digital (28, 66, 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]

(198−37, 198, 11319)-Net over F₄ — Digital

Digital (161, 198, 11319)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁹⁸, 11319, F₄, 37) (dual of [11319, 11121, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁹⁸, 16400, F₄, 37) (dual of [16400, 16202, 38]-code), using
  - constructionÂ X applied to C([0,18]) âŠ‚ C([0,17]) [i] based on
    1. linear OA(4¹⁹⁷, 16385, F₄, 37) (dual of [16385, 16188, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (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(4¹, 15, F₄, 1) (dual of [15, 14, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(198−37, 198, large)-Net in Base 4 — Upper bound on s

There is no (161, 198, large)-net in base 4, because

35 times m-reduction [i] would yield (161, 163, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]