MinT - Best Known (29−14, 29, s)-Nets in Base 25

Best Known (29−14, 29, s)-Nets in Base 25

(29−14, 29, 132)-Net over F₂₅ — Constructive and digital

Digital (15, 29, 132)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 11, 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]
2. digital (4, 18, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅ (see above)

(29−14, 29, 398)-Net over F₂₅ — Digital

Digital (15, 29, 398)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25²⁹, 398, F₂₅, 14) (dual of [398, 369, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(25²⁹, 633, F₂₅, 14) (dual of [633, 604, 15]-code), using
  - constructionÂ X applied to C_e(13) âŠ‚ C_e(10) [i] based on
    1. linear OA(25²⁷, 625, F₂₅, 14) (dual of [625, 598, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
    2. linear OA(25²¹, 625, F₂₅, 11) (dual of [625, 604, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    3. linear OA(25², 8, F₂₅, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,25)), using
      - discarding factors / shortening the dual code based on linear OA(25², 25, F₂₅, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
        Reedâ€“Solomon code RS(23,25) [i]

(29−14, 29, 87127)-Net in Base 25 — Upper bound on s

There is no (15, 29, 87128)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 34695 026212 631899 136104 035097 750650 949825 > 25²⁹ [i]