MinT - Best Known (34−11, 34, s)-Nets in Base 5

Best Known (34−11, 34, s)-Nets in Base 5

(34−11, 34, 132)-Net over F₅ — Constructive and digital

Digital (23, 34, 132)-net over F₅, using

4 times m-reduction [i] based on digital (23, 38, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 19, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
      - a function field by GarcÃa and Quoos [i]

(34−11, 34, 373)-Net over F₅ — Digital

Digital (23, 34, 373)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5³⁴, 373, F₅, 11) (dual of [373, 339, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(5³⁴, 630, F₅, 11) (dual of [630, 596, 12]-code), using
  - constructionÂ X applied to C_e(10) âŠ‚ C_e(8) [i] based on
    1. linear OA(5³³, 625, F₅, 11) (dual of [625, 592, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    2. linear OA(5²⁹, 625, F₅, 9) (dual of [625, 596, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
    3. linear OA(5¹, 5, F₅, 1) (dual of [5, 4, 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]

(34−11, 34, 26725)-Net in Base 5 — Upper bound on s

There is no (23, 34, 26726)-net in base 5, because

1 times m-reduction [i] would yield (23, 33, 26726)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 116426 693308 764628 378905 > 5³³ [i]