MinT - Best Known (167, 206, s)-Nets in Base 4

Best Known (167, 206, s)-Nets in Base 4

(167, 206, 1539)-Net over F₄ — Constructive and digital

Digital (167, 206, 1539)-net over F₄, using

t-expansion [i] based on digital (166, 206, 1539)-net over F₄, using
- 1 times m-reduction [i] based on digital (166, 207, 1539)-net over F₄, using
  - trace code for nets [i] based on digital (28, 69, 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]

(167, 206, 10551)-Net over F₄ — Digital

Digital (167, 206, 10551)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁰⁶, 10551, F₄, 39) (dual of [10551, 10345, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁰⁶, 16400, F₄, 39) (dual of [16400, 16194, 40]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(4²⁰⁵, 16399, F₄, 39) (dual of [16399, 16194, 40]-code), using
    - constructionÂ X applied to C_e(38) âŠ‚ C_e(36) [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(4¹⁹⁰, 16384, F₄, 37) (dual of [16384, 16194, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [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]

(167, 206, 8279495)-Net in Base 4 — Upper bound on s

There is no (167, 206, 8279496)-net in base 4, because

1 times m-reduction [i] would yield (167, 205, 8279496)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2644 226252 622329 050895 307608 331950 319122 826999 887769 483198 491497 832930 206536 003584 605184 427886 937656 169700 573302 501077 300036 > 4²⁰⁵ [i]