MinT - Best Known (141−36, 141, s)-Nets in Base 5

Best Known (141−36, 141, s)-Nets in Base 5

(141−36, 141, 410)-Net over F₅ — Constructive and digital

Digital (105, 141, 410)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (1, 19, 10)-net over F₅, using
  - net from sequence [i] based on digital (1, 9)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 1 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (86, 122, 400)-net over F₅, using
  - trace code for nets [i] based on digital (25, 61, 200)-net over F₂₅, using
    - net from sequence [i] based on digital (25, 199)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 25 and N(F) ≥ 200, using
        a function field by GarcÃa and Quoos [i]

(141−36, 141, 2532)-Net over F₅ — Digital

Digital (105, 141, 2532)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁴¹, 2532, F₅, 36) (dual of [2532, 2391, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(5¹⁴¹, 3125, F₅, 36) (dual of [3125, 2984, 37]-code), using
  - an extension C_e(35) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [i]

(141−36, 141, 564059)-Net in Base 5 — Upper bound on s

There is no (105, 141, 564060)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 358 735571 103974 995413 687234 622849 236663 959498 545264 411569 974696 667241 901156 584552 408416 913732 958785 > 5¹⁴¹ [i]