MinT - Best Known (13, 13+10, s)-Nets in Base 5

Best Known (13, 13+10, s)-Nets in Base 5

Digital (13, 23, 54)-net over F₅, using

1 times m-reduction [i] based on digital (13, 24, 54)-net over F₅, using
- trace code for nets [i] based on digital (1, 12, 27)-net over F₂₅, using
  - net from sequence [i] based on digital (1, 26)-sequence over F₂₅, using
    - Niederreiter sequence [i]

Digital (13, 23, 75)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5²³, 75, F₅, 10) (dual of [75, 52, 11]-code), using
- 2 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 0) [i] based on linear OA(5²², 72, F₅, 10) (dual of [72, 50, 11]-code), using
  - trace code [i] based on linear OA(25¹¹, 36, F₂₅, 10) (dual of [36, 25, 11]-code), using
    - extended algebraic-geometric code AG_e(F,25P) [i] based on function field F/F₂₅ with g(F) = 1 and N(F) ≥ 36, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

There is no (13, 23, 1066)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 11926 339798 381737 > 5²³ [i]