MinT - Best Known (43−10, 43, s)-Nets in Base 4

Best Known (43−10, 43, s)-Nets in Base 4

(43−10, 43, 1028)-Net over F₄ — Constructive and digital

Digital (33, 43, 1028)-net over F₄, using

1 times m-reduction [i] based on digital (33, 44, 1028)-net over F₄, using
- trace code for nets [i] based on digital (0, 11, 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]

(43−10, 43, 2051)-Net over F₄ — Digital

Digital (33, 43, 2051)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4⁴³, 2051, F₄, 2, 10) (dual of [(2051, 2), 4059, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4⁴³, 4102, F₄, 10) (dual of [4102, 4059, 11]-code), using
  - constructionÂ X applied to C_e(9) âŠ‚ C_e(8) [i] based on
    1. linear OA(4⁴³, 4096, F₄, 10) (dual of [4096, 4053, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
    2. linear OA(4³⁷, 4096, F₄, 9) (dual of [4096, 4059, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
    3. linear OA(4⁰, 6, F₄, 0) (dual of [6, 6, 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]

(43−10, 43, 130743)-Net in Base 4 — Upper bound on s

There is no (33, 43, 130744)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 77 373948 955642 346988 145303 > 4⁴³ [i]