MinT - Best Known (89, 119, s)-Nets in Base 4

Best Known (89, 119, s)-Nets in Base 4

(89, 119, 531)-Net over F₄ — Constructive and digital

Digital (89, 119, 531)-net over F₄, using

4 times m-reduction [i] based on digital (89, 123, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 41, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(89, 119, 576)-Net in Base 4 — Constructive

(89, 119, 576)-net in base 4, using

1 times m-reduction [i] based on (89, 120, 576)-net in base 4, using
- trace code for nets [i] based on (9, 40, 192)-net in base 64, using
  - 2 times m-reduction [i] based on (9, 42, 192)-net in base 64, using
    - base change [i] based on digital (3, 36, 192)-net over F₁₂₈, using
      - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(89, 119, 1169)-Net over F₄ — Digital

Digital (89, 119, 1169)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹¹⁹, 1169, F₄, 30) (dual of [1169, 1050, 31]-code), using
- 132 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) [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]

(89, 119, 127922)-Net in Base 4 — Upper bound on s

There is no (89, 119, 127923)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 441732 943651 614223 713864 262760 719078 596649 237607 121197 937333 658614 181696 > 4¹¹⁹ [i]