MinT - Best Known (38−13, 38, s)-Nets in Base 4

Best Known (38−13, 38, s)-Nets in Base 4

(38−13, 38, 130)-Net over F₄ — Constructive and digital

Digital (25, 38, 130)-net over F₄, using

trace code for nets [i] based on digital (6, 19, 65)-net over F₁₆, using
- net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
    - the Hermitian function field over F₁₆ [i]

(38−13, 38, 166)-Net over F₄ — Digital

Digital (25, 38, 166)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4³⁸, 166, F₄, 13) (dual of [166, 128, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4³⁸, 261, F₄, 13) (dual of [261, 223, 14]-code), using
  - constructionÂ X applied to C_e(12) âŠ‚ C_e(10) [i] based on
    1. linear OA(4³⁷, 256, F₄, 13) (dual of [256, 219, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    2. linear OA(4³³, 256, F₄, 11) (dual of [256, 223, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    3. linear OA(4¹, 5, F₄, 1) (dual of [5, 4, 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]

(38−13, 38, 5145)-Net in Base 4 — Upper bound on s

There is no (25, 38, 5146)-net in base 4, because

1 times m-reduction [i] would yield (25, 37, 5146)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 18901 252467 851950 815004 > 4³⁷ [i]