MinT - Best Known (34, 67, s)-Nets in Base 25

Best Known (34, 67, s)-Nets in Base 25

(34, 67, 208)-Net over F₂₅ — Constructive and digital

Digital (34, 67, 208)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (9, 25, 104)-net over F₂₅, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by GarcÃa and Quoos [i]
2. digital (9, 42, 104)-net over F₂₅, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₂₅ (see above)

(34, 67, 476)-Net over F₂₅ — Digital

Digital (34, 67, 476)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁶⁷, 476, F₂₅, 33) (dual of [476, 409, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(25⁶⁷, 642, F₂₅, 33) (dual of [642, 575, 34]-code), using
  - constructionÂ X applied to C_e(32) âŠ‚ C_e(26) [i] based on
    1. linear OA(25⁶², 625, F₂₅, 33) (dual of [625, 563, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]
    2. linear OA(25⁵⁰, 625, F₂₅, 27) (dual of [625, 575, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
    3. linear OA(25⁵, 17, F₂₅, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,25)), using
      - discarding factors / shortening the dual code based on linear OA(25⁵, 25, F₂₅, 5) (dual of [25, 20, 6]-code or 25-arc in PG(4,25)), using
        Reedâ€“Solomon code RS(20,25) [i]

(34, 67, 165504)-Net in Base 25 — Upper bound on s

There is no (34, 67, 165505)-net in base 25, because

1 times m-reduction [i] would yield (34, 66, 165505)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 183 688105 641593 231187 773169 542736 109230 126133 438596 404101 214204 191942 937892 953907 587107 991425 > 25⁶⁶ [i]