MinT - Best Known (25, 25+10, s)-Nets in Base 5

Best Known (25, 25+10, s)-Nets in Base 5

(25, 25+10, 156)-Net over F₅ — Constructive and digital

Digital (25, 35, 156)-net over F₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (2, 5, 52)-net over F₅, using
  - s-reduction based on digital (2, 5, 66)-net over F₅, using
    - net defined by OOA [i] based on linear OOA(5⁵, 66, F₅, 3, 3) (dual of [(66, 3), 193, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(5⁵, 66, F₅, 2, 3) (dual of [(66, 2), 127, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(5⁵, 66, F₅, 3) (dual of [66, 61, 4]-code or 66-cap in PG(4,5)), using
        caps completed using a computer search [i]
2. digital (5, 10, 52)-net over F₅, using
  - s-reduction based on digital (5, 10, 68)-net over F₅, using
    - linear code embedding found in a computer search [i]
3. digital (10, 20, 52)-net over F₅, using
  - trace code for nets [i] based on digital (0, 10, 26)-net over F₂₅, using
    - net from sequence [i] based on digital (0, 25)-sequence over F₂₅, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 0 and N(F) ≥ 26, using
        the rational function field F₂₅(x) [i]
      - Niederreiter sequence [i]

(25, 25+10, 650)-Net over F₅ — Digital

Digital (25, 35, 650)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5³⁵, 650, F₅, 10) (dual of [650, 615, 11]-code), using
- 22 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 17 times 0) [i] based on linear OA(5³², 625, F₅, 10) (dual of [625, 593, 11]-code), using
  - 1 times truncation [i] based on linear OA(5³³, 626, F₅, 11) (dual of [626, 593, 12]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 626Â | 5⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(25, 25+10, 50878)-Net in Base 5 — Upper bound on s

There is no (25, 35, 50879)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2 910383 666544 050157 849325 > 5³⁵ [i]