MinT - Best Known (190−35, 190, s)-Nets in Base 4

Best Known (190−35, 190, s)-Nets in Base 4

(190−35, 190, 1539)-Net over F₄ — Constructive and digital

Digital (155, 190, 1539)-net over F₄, using

4¹ times duplication [i] based on digital (154, 189, 1539)-net over F₄, using
- trace code for nets [i] based on digital (28, 63, 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]

(190−35, 190, 12288)-Net over F₄ — Digital

Digital (155, 190, 12288)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁹⁰, 12288, F₄, 35) (dual of [12288, 12098, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁹⁰, 16419, F₄, 35) (dual of [16419, 16229, 36]-code), using
  - constructionÂ X applied to C_e(34) âŠ‚ C_e(29) [i] based on
    1. 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]
    2. linear OA(4¹⁵⁵, 16384, F₄, 30) (dual of [16384, 16229, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
    3. linear OA(4⁷, 35, F₄, 4) (dual of [35, 28, 5]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁷, 43, F₄, 4) (dual of [43, 36, 5]-code), using
        quaternary constacyclic codes with minimum distance 5 [i]

(190−35, 190, large)-Net in Base 4 — Upper bound on s

There is no (155, 190, large)-net in base 4, because

33 times m-reduction [i] would yield (155, 157, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]