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

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

(34, 34+34, 204)-Net over F₂₅ — Constructive and digital

Digital (34, 68, 204)-net over F₂₅, using

t-expansion [i] based on digital (30, 68, 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]

(34, 34+34, 436)-Net over F₂₅ — Digital

Digital (34, 68, 436)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁶⁸, 436, F₂₅, 34) (dual of [436, 368, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(25⁶⁸, 639, F₂₅, 34) (dual of [639, 571, 35]-code), using
  - constructionÂ X applied to C_e(33) âŠ‚ C_e(28) [i] based on
    1. linear OA(25⁶⁴, 625, F₂₅, 34) (dual of [625, 561, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    2. linear OA(25⁵⁴, 625, F₂₅, 29) (dual of [625, 571, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    3. linear OA(25⁴, 14, F₂₅, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,25)), using
      - discarding factors / shortening the dual code based on linear OA(25⁴, 25, F₂₅, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
        Reedâ€“Solomon code RS(21,25) [i]

(34, 34+34, 116805)-Net in Base 25 — Upper bound on s

There is no (34, 68, 116806)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 114810 227236 089786 803064 022702 619580 501960 485940 598119 776666 917172 494078 049862 815559 927380 609425 > 25⁶⁸ [i]