MinT - Best Known (92, 92+30, s)-Nets in Base 5

Best Known (92, 92+30, s)-Nets in Base 5

(92, 92+30, 408)-Net over F₅ — Constructive and digital

Digital (92, 122, 408)-net over F₅, using

2 times m-reduction [i] based on digital (92, 124, 408)-net over F₅, using
- trace code for nets [i] based on digital (30, 62, 204)-net over F₂₅, using
  - net from sequence [i] based on digital (30, 203)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 30 and N(F) ≥ 204, using
      - a function field by GarcÃa and Quoos [i]

(92, 92+30, 2942)-Net over F₅ — Digital

Digital (92, 122, 2942)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹²², 2942, F₅, 30) (dual of [2942, 2820, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(5¹²², 3136, F₅, 30) (dual of [3136, 3014, 31]-code), using
  - constructionÂ X applied to C_e(30) âŠ‚ C_e(27) [i] based on
    1. linear OA(5¹²¹, 3125, F₅, 31) (dual of [3125, 3004, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
    2. linear OA(5¹¹¹, 3125, F₅, 28) (dual of [3125, 3014, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    3. linear OA(5¹, 11, F₅, 1) (dual of [11, 10, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(5¹, s, F₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(92, 92+30, 777421)-Net in Base 5 — Upper bound on s

There is no (92, 122, 777422)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 18 808005 095844 047480 765902 558251 984160 128907 666745 179212 948712 510507 639539 795144 040281 > 5¹²² [i]