MinT - Best Known (64−32, 64, s)-Nets in Base 25

Best Known (64−32, 64, s)-Nets in Base 25

(64−32, 64, 204)-Net over F₂₅ — Constructive and digital

Digital (32, 64, 204)-net over F₂₅, using

t-expansion [i] based on digital (30, 64, 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]

(64−32, 64, 419)-Net over F₂₅ — Digital

Digital (32, 64, 419)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁶⁴, 419, F₂₅, 32) (dual of [419, 355, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(25⁶⁴, 639, F₂₅, 32) (dual of [639, 575, 33]-code), using
  - constructionÂ X applied to C_e(31) âŠ‚ C_e(26) [i] based on
    1. linear OA(25⁶⁰, 625, F₂₅, 32) (dual of [625, 565, 33]-code), using an extension C_e(31) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,31], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [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(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]

(64−32, 64, 110676)-Net in Base 25 — Upper bound on s

There is no (32, 64, 110677)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 293894 215472 190676 056832 777587 318958 855057 517472 850188 577142 642738 178773 765289 036602 818945 > 25⁶⁴ [i]