MinT - Best Known (30, 40, s)-Nets in Base 4

Best Known (30, 40, s)-Nets in Base 4

(30, 40, 1028)-Net over F₄ — Constructive and digital

Digital (30, 40, 1028)-net over F₄, using

trace code for nets [i] based on digital (0, 10, 257)-net over F₂₅₆, using
- net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
    - the rational function field F₂₅₆(x) [i]
  - Niederreiter sequence [i]

(30, 40, 1050)-Net over F₄ — Digital

Digital (30, 40, 1050)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁴⁰, 1050, F₄, 10) (dual of [1050, 1010, 11]-code), using
- 17 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 4 times 0, 1, 9 times 0) [i] based on linear OA(4³⁶, 1029, F₄, 10) (dual of [1029, 993, 11]-code), using
  - constructionÂ X applied to C_e(9) âŠ‚ C_e(8) [i] based on
    1. linear OA(4³⁶, 1024, F₄, 10) (dual of [1024, 988, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
    2. linear OA(4³¹, 1024, F₄, 9) (dual of [1024, 993, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
    3. linear OA(4⁰, 5, F₄, 0) (dual of [5, 5, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(30, 40, 56907)-Net in Base 4 — Upper bound on s

There is no (30, 40, 56908)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 209014 598209 071026 634882 > 4⁴⁰ [i]