MinT - Best Known (87−51, 87, s)-Nets in Base 25

Best Known (87−51, 87, s)-Nets in Base 25

(87−51, 87, 208)-Net over F₂₅ — Constructive and digital

Digital (36, 87, 208)-net over F₂₅, using

t-expansion [i] based on digital (35, 87, 208)-net over F₂₅, using
- net from sequence [i] based on digital (35, 207)-sequence over F₂₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 35 and N(F) ≥ 208, using
    - a function field by GarcÃa and Quoos [i]

(87−51, 87, 210)-Net over F₂₅ — Digital

Digital (36, 87, 210)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(25⁸⁷, 210, F₂₅, 2, 51) (dual of [(210, 2), 333, 52]-NRT-code), using
- constructionÂ X applied to AG(2;F,346P) âŠ‚ AG(2;F,358P) [i] based on
  1. linear OOA(25⁷⁶, 199, F₂₅, 2, 51) (dual of [(199, 2), 322, 52]-NRT-code), using algebraic-geometric NRT-code AG(2;F,346P) [i] based on function field F/F₂₅ with g(F) = 25 and N(F) ≥ 200, using
    - a function field by GarcÃa and Quoos [i]
  2. linear OOA(25⁶⁴, 199, F₂₅, 2, 39) (dual of [(199, 2), 334, 40]-NRT-code), using algebraic-geometric NRT-code AG(2;F,358P) [i] based on function field F/F₂₅ with g(F) = 25 and N(F) ≥ 200 (see above)
  3. linear OOA(25¹¹, 11, F₂₅, 2, 11) (dual of [(11, 2), 11, 12]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(25¹¹, 25, F₂₅, 2, 11) (dual of [(25, 2), 39, 12]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(2;39,25) [i]

(87−51, 87, 27297)-Net in Base 25 — Upper bound on s

There is no (36, 87, 27298)-net in base 25, because

1 times m-reduction [i] would yield (36, 86, 27298)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 1 671442 406542 945518 642808 789225 566530 546921 113034 085581 365660 633087 592292 060354 394781 795298 000624 229472 108516 765362 373809 > 25⁸⁶ [i]