MinT - Best Known (90, 120, s)-Nets in Base 4

Best Known (90, 120, s)-Nets in Base 4

(90, 120, 1028)-Net over F₄ — Constructive and digital

Digital (90, 120, 1028)-net over F₄, using

trace code for nets [i] based on digital (0, 30, 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]

(90, 120, 1222)-Net over F₄ — Digital

Digital (90, 120, 1222)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹²⁰, 1222, F₄, 30) (dual of [1222, 1102, 31]-code), using
- 184 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 13 times 0, 1, 23 times 0, 1, 35 times 0, 1, 44 times 0, 1, 51 times 0) [i] based on linear OA(4¹¹¹, 1029, F₄, 30) (dual of [1029, 918, 31]-code), using
  - constructionÂ X applied to C_e(29) âŠ‚ C_e(28) [i] based on
    1. linear OA(4¹¹¹, 1024, F₄, 30) (dual of [1024, 913, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
    2. linear OA(4¹⁰⁶, 1024, F₄, 29) (dual of [1024, 918, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [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]

(90, 120, 140309)-Net in Base 4 — Upper bound on s

There is no (90, 120, 140310)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 766874 413550 910465 332970 450890 849187 466220 280636 047005 594441 364113 970088 > 4¹²⁰ [i]