MinT - Best Known (247−47, 247, s)-Nets in Base 4

Best Known (247−47, 247, s)-Nets in Base 4

(247−47, 247, 1539)-Net over F₄ — Constructive and digital

Digital (200, 247, 1539)-net over F₄, using

11 times m-reduction [i] based on digital (200, 258, 1539)-net over F₄, using
- trace code for nets [i] based on digital (28, 86, 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]

(247−47, 247, 11454)-Net over F₄ — Digital

Digital (200, 247, 11454)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁴⁷, 11454, F₄, 47) (dual of [11454, 11207, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁴⁷, 16399, F₄, 47) (dual of [16399, 16152, 48]-code), using
  - constructionÂ X applied to C_e(46) âŠ‚ C_e(44) [i] based on
    1. linear OA(4²⁴⁶, 16384, F₄, 47) (dual of [16384, 16138, 48]-code), using an extension C_e(46) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,46], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 47 [i]
    2. linear OA(4²³², 16384, F₄, 45) (dual of [16384, 16152, 46]-code), using an extension C_e(44) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,44], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 45 [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]

(247−47, 247, large)-Net in Base 4 — Upper bound on s

There is no (200, 247, large)-net in base 4, because

45 times m-reduction [i] would yield (200, 202, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]