MinT - Best Known (35−16, 35, s)-Nets in Base 25

Best Known (35−16, 35, s)-Nets in Base 25

(35−16, 35, 153)-Net over F₂₅ — Constructive and digital

Digital (19, 35, 153)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 27)-net over F₂₅, using
  - net from sequence [i] based on digital (1, 26)-sequence over F₂₅, using
    - Niederreiter sequence [i]
2. digital (10, 26, 126)-net over F₂₅, using
  - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
      - the Hermitian function field over F₂₅ [i]

(35−16, 35, 619)-Net over F₂₅ — Digital

Digital (19, 35, 619)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25³⁵, 619, F₂₅, 16) (dual of [619, 584, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(25³⁵, 639, F₂₅, 16) (dual of [639, 604, 17]-code), using
  - constructionÂ X applied to C_e(15) âŠ‚ C_e(10) [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(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(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]

(35−16, 35, 204860)-Net in Base 25 — Upper bound on s

There is no (19, 35, 204861)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 8 470493 776533 890514 154690 048795 525200 098282 814145 > 25³⁵ [i]