MinT - Best Known (218−39, 218, s)-Nets in Base 4

Best Known (218−39, 218, s)-Nets in Base 4

(218−39, 218, 1539)-Net over F₄ — Constructive and digital

Digital (179, 218, 1539)-net over F₄, using

t-expansion [i] based on 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]

(218−39, 218, 16440)-Net over F₄ — Digital

Digital (179, 218, 16440)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²¹⁸, 16440, F₄, 39) (dual of [16440, 16222, 40]-code), using
- constructionÂ X applied to C_e(38) âŠ‚ C_e(30) [i] based on
  1. linear OA(4²⁰⁴, 16384, F₄, 39) (dual of [16384, 16180, 40]-code), using an extension C_e(38) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]
  2. linear OA(4¹⁶², 16384, F₄, 31) (dual of [16384, 16222, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
  3. linear OA(4¹⁴, 56, F₄, 7) (dual of [56, 42, 8]-code), using
    - discarding factors / shortening the dual code based on linear OA(4¹⁴, 65, F₄, 7) (dual of [65, 51, 8]-code), using
      - a â€œGraXXâ€ code from Grasslâ€™s database [i]

(218−39, 218, large)-Net in Base 4 — Upper bound on s

There is no (179, 218, large)-net in base 4, because

37 times m-reduction [i] would yield (179, 181, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]