MinT - Best Known (25, 25+11, s)-Nets in Base 5

Best Known (25, 25+11, s)-Nets in Base 5

(25, 25+11, 144)-Net over F₅ — Constructive and digital

Digital (25, 36, 144)-net over F₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (2, 5, 46)-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 (4, 9, 46)-net over F₅, using
  - linear code embedding found in a computer search [i]
3. digital (11, 22, 52)-net over F₅, using
  - trace code for nets [i] based on digital (0, 11, 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+11, 536)-Net over F₅ — Digital

Digital (25, 36, 536)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5³⁶, 536, F₅, 11) (dual of [536, 500, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(5³⁶, 624, F₅, 11) (dual of [624, 588, 12]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(25, 25+11, 50878)-Net in Base 5 — Upper bound on s

There is no (25, 36, 50879)-net in base 5, because

1 times m-reduction [i] would yield (25, 35, 50879)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2 910383 666544 050157 849325 > 5³⁵ [i]