MinT - Best Known (20−8, 20, s)-Nets in Base 8

Best Known (20−8, 20, s)-Nets in Base 8

Digital (12, 20, 160)-net over F₈, using

2 times m-reduction [i] based on digital (12, 22, 160)-net over F₈, using
- trace code for nets [i] based on digital (1, 11, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

Digital (12, 20, 196)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8²⁰, 196, F₈, 8) (dual of [196, 176, 9]-code), using
- 32 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 5 times 0, 1, 25 times 0) [i] based on linear OA(8¹⁸, 162, F₈, 8) (dual of [162, 144, 9]-code), using
  - trace code [i] based on linear OA(64⁹, 81, F₆₄, 8) (dual of [81, 72, 9]-code), using
    - extended algebraic-geometric code AG_e(F,72P) [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 81, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(12, 20, 258)-net in base 8, using

There is no (12, 20, 10360)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 153366 197441 188431 > 8²⁰ [i]