MinT - Best Known (27, 58, s)-Nets in Base 5

Best Known (27, 58, s)-Nets in Base 5

(27, 58, 51)-Net over F₅ — Constructive and digital

Digital (27, 58, 51)-net over F₅, using

t-expansion [i] based on digital (22, 58, 51)-net over F₅, using
- net from sequence [i] based on digital (22, 50)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 22 and N(F) ≥ 51, using
    - an explicitly constructive algebraic function field [i]

(27, 58, 56)-Net over F₅ — Digital

Digital (27, 58, 56)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(5⁵⁸, 56, F₅, 3, 31) (dual of [(56, 3), 110, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(5⁵⁸, 57, F₅, 3, 31) (dual of [(57, 3), 113, 32]-NRT-code), using
  - constructionÂ X applied to AG(3;F,130P) âŠ‚ AG(3;F,135P) [i] based on
    1. linear OOA(5⁵⁴, 54, F₅, 3, 31) (dual of [(54, 3), 108, 32]-NRT-code), using algebraic-geometric NRT-code AG(3;F,130P) [i] based on function field F/F₅ with g(F) = 23 and N(F) ≥ 55, using
      - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]
    2. linear OOA(5⁴⁹, 54, F₅, 3, 26) (dual of [(54, 3), 113, 27]-NRT-code), using algebraic-geometric NRT-code AG(3;F,135P) [i] based on function field F/F₅ with g(F) = 23 and N(F) ≥ 55 (see above)
    3. linear OOA(5⁴, 3, F₅, 3, 4) (dual of [(3, 3), 5, 5]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(5⁴, 5, F₅, 3, 4) (dual of [(5, 3), 11, 5]-NRT-code), using
        Reedâ€“Solomon NRT-code RS(3;11,5) [i]

(27, 58, 716)-Net in Base 5 — Upper bound on s

There is no (27, 58, 717)-net in base 5, because

1 times m-reduction [i] would yield (27, 57, 717)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 6976 194305 124193 771845 264077 893481 062269 > 5⁵⁷ [i]