MinT - Best Known (105, 105+38, s)-Nets in Base 4

Best Known (105, 105+38, s)-Nets in Base 4

(105, 105+38, 531)-Net over F₄ — Constructive and digital

Digital (105, 143, 531)-net over F₄, using

4 times m-reduction [i] based on digital (105, 147, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 49, 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]

(105, 105+38, 1063)-Net over F₄ — Digital

Digital (105, 143, 1063)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁴³, 1063, F₄, 38) (dual of [1063, 920, 39]-code), using
- 32 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 9 times 0, 1, 21 times 0) [i] based on linear OA(4¹⁴¹, 1029, F₄, 38) (dual of [1029, 888, 39]-code), using
  - constructionÂ X applied to C_e(37) âŠ‚ C_e(36) [i] based on
    1. linear OA(4¹⁴¹, 1024, F₄, 38) (dual of [1024, 883, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
    2. linear OA(4¹³⁶, 1024, F₄, 37) (dual of [1024, 888, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [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]

(105, 105+38, 89807)-Net in Base 4 — Upper bound on s

There is no (105, 143, 89808)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 124 337837 735756 768433 553542 767740 824281 758137 972951 106435 596149 643417 526366 095047 427628 > 4¹⁴³ [i]