MinT - Best Known (219, 219+35, s)-Nets in Base 4

Best Known (219, 219+35, s)-Nets in Base 4

(219, 219+35, 15430)-Net over F₄ — Constructive and digital

Digital (219, 254, 15430)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 19, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (200, 235, 15420)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4²³⁵, 15420, F₄, 35, 35) (dual of [(15420, 35), 539465, 36]-NRT-code), using
    - OOA 17-folding and stacking with additional row [i] based on linear OA(4²³⁵, 262141, F₄, 35) (dual of [262141, 261906, 36]-code), using
      - discarding factors / shortening the dual code based on linear OA(4²³⁵, 262144, F₄, 35) (dual of [262144, 261909, 36]-code), using
        an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 4⁹âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]

(219, 219+35, 181118)-Net over F₄ — Digital

Digital (219, 254, 181118)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁵⁴, 181118, F₄, 35) (dual of [181118, 180864, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁵⁴, 262183, F₄, 35) (dual of [262183, 261929, 36]-code), using
  - constructionÂ X4 applied to C([0,18]) âŠ‚ C([0,16]) [i] based on
    1. linear OA(4²⁵³, 262145, F₄, 37) (dual of [262145, 261892, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 4¹⁸âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
    2. linear OA(4²¹⁷, 262145, F₄, 33) (dual of [262145, 261928, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 4¹⁸âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
    3. linear OA(4³⁷, 38, F₄, 37) (dual of [38, 1, 38]-code or 38-arc in PG(36,4)), using
      - dual of repetition code with length 38 [i]
    4. linear OA(4¹, 38, F₄, 1) (dual of [38, 37, 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]

(219, 219+35, large)-Net in Base 4 — Upper bound on s

There is no (219, 254, large)-net in base 4, because

33 times m-reduction [i] would yield (219, 221, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]