MinT - Best Known (75, 101, s)-Nets in Base 4

Best Known (75, 101, s)-Nets in Base 4

(75, 101, 531)-Net over F₄ — Constructive and digital

Digital (75, 101, 531)-net over F₄, using

1 times m-reduction [i] based on digital (75, 102, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 34, 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]

(75, 101, 1036)-Net over F₄ — Digital

Digital (75, 101, 1036)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁰¹, 1036, F₄, 26) (dual of [1036, 935, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁰¹, 1044, F₄, 26) (dual of [1044, 943, 27]-code), using
  - constructionÂ X applied to C_e(25) âŠ‚ C_e(21) [i] based on
    1. linear OA(4⁹⁶, 1024, F₄, 26) (dual of [1024, 928, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
    2. linear OA(4⁸¹, 1024, F₄, 22) (dual of [1024, 943, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    3. linear OA(4⁵, 20, F₄, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,4)), using
      - discarding factors / shortening the dual code based on linear OA(4⁵, 41, F₄, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
        a cap found by Tallinii [i]

(75, 101, 89903)-Net in Base 4 — Upper bound on s

There is no (75, 101, 89904)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6 427943 299843 177404 112345 221604 449460 846162 447758 350485 025827 > 4¹⁰¹ [i]