MinT - Best Known (62−30, 62, s)-Nets in Base 25

Best Known (62−30, 62, s)-Nets in Base 25

(62−30, 62, 204)-Net over F₂₅ — Constructive and digital

Digital (32, 62, 204)-net over F₂₅, using

t-expansion [i] based on digital (30, 62, 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]

(62−30, 62, 510)-Net over F₂₅ — Digital

Digital (32, 62, 510)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁶², 510, F₂₅, 30) (dual of [510, 448, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(25⁶², 642, F₂₅, 30) (dual of [642, 580, 31]-code), using
  - constructionÂ X applied to C_e(29) âŠ‚ C_e(22) [i] based on
    1. linear OA(25⁵⁶, 625, F₂₅, 30) (dual of [625, 569, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
    2. linear OA(25⁴⁵, 625, F₂₅, 23) (dual of [625, 580, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    3. linear OA(25⁶, 17, F₂₅, 6) (dual of [17, 11, 7]-code or 17-arc in PG(5,25)), using
      - discarding factors / shortening the dual code based on linear OA(25⁶, 25, F₂₅, 6) (dual of [25, 19, 7]-code or 25-arc in PG(5,25)), using
        Reedâ€“Solomon code RS(19,25) [i]

(62−30, 62, 160578)-Net in Base 25 — Upper bound on s

There is no (32, 62, 160579)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 470 229176 544694 048140 036200 952740 033152 115282 154163 873363 032719 780600 072923 387090 185145 > 25⁶² [i]