MinT - Best Known (209−39, 209, s)-Nets in Base 4

Best Known (209−39, 209, s)-Nets in Base 4

(209−39, 209, 1539)-Net over F₄ — Constructive and digital

Digital (170, 209, 1539)-net over F₄, using

4 times m-reduction [i] based on digital (170, 213, 1539)-net over F₄, using
- trace code for nets [i] based on digital (28, 71, 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]

(209−39, 209, 11810)-Net over F₄ — Digital

Digital (170, 209, 11810)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁰⁹, 11810, F₄, 39) (dual of [11810, 11601, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁰⁹, 16410, F₄, 39) (dual of [16410, 16201, 40]-code), using
  - constructionÂ X applied to C_e(38) âŠ‚ C_e(34) [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(4¹⁸³, 16384, F₄, 35) (dual of [16384, 16201, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
    3. linear OA(4⁵, 26, F₄, 3) (dual of [26, 21, 4]-code or 26-cap in PG(4,4)), using
      - discarding factors / shortening the dual code based on linear OA(4⁵, 41, F₄, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
        a cap found by Tallinii [i]

(209−39, 209, large)-Net in Base 4 — Upper bound on s

There is no (170, 209, large)-net in base 4, because

37 times m-reduction [i] would yield (170, 172, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]