MinT - Best Known (106, 106+35, s)-Nets in Base 5

Best Known (106, 106+35, s)-Nets in Base 5

(106, 106+35, 418)-Net over F₅ — Constructive and digital

Digital (106, 141, 418)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 21, 18)-net over F₅, using
  - net from sequence [i] based on digital (4, 17)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 4 and N(F) ≥ 18, using
      - an explicitly constructive algebraic function field [i]
2. digital (85, 120, 400)-net over F₅, using
  - trace code for nets [i] based on digital (25, 60, 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]

(106, 106+35, 3015)-Net over F₅ — Digital

Digital (106, 141, 3015)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁴¹, 3015, F₅, 35) (dual of [3015, 2874, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(5¹⁴¹, 3126, F₅, 35) (dual of [3126, 2985, 36]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 3126Â | 5¹⁰âˆ’1, defining interval IÂ =Â [0,17], and minimum distance dÂ â‰¥ |{−17,−16,…,17}|+1Â =Â 36 (BCH-bound) [i]

(106, 106+35, 1023537)-Net in Base 5 — Upper bound on s

There is no (106, 141, 1023538)-net in base 5, because

1 times m-reduction [i] would yield (106, 140, 1023538)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 71 747571 370521 409205 446279 234064 085732 440696 860402 926220 615187 592864 830673 864226 804097 684893 413065 > 5¹⁴⁰ [i]