MinT - Best Known (32−16, 32, s)-Nets in Base 25

Best Known (32−16, 32, s)-Nets in Base 25

(32−16, 32, 132)-Net over F₂₅ — Constructive and digital

Digital (16, 32, 132)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 12, 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, 20, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅ (see above)

(32−16, 32, 315)-Net over F₂₅ — Digital

Digital (16, 32, 315)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(25³², 315, F₂₅, 2, 16) (dual of [(315, 2), 598, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25³², 630, F₂₅, 16) (dual of [630, 598, 17]-code), using
  - constructionÂ X applied to C_e(15) âŠ‚ C_e(13) [i] based on
    1. linear OA(25³¹, 625, F₂₅, 16) (dual of [625, 594, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    2. 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]
    3. linear OA(25¹, 5, F₂₅, 1) (dual of [5, 4, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(25¹, s, F₂₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(32−16, 32, 61264)-Net in Base 25 — Upper bound on s

There is no (16, 32, 61265)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 542 108593 943507 972029 664655 414861 798198 178753 > 25³² [i]