MinT - Best Known (21−9, 21, s)-Nets in Base 8

Best Known (21−9, 21, s)-Nets in Base 8

Digital (12, 21, 160)-net over F₈, using

1 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, 21, 165)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8²¹, 165, F₈, 9) (dual of [165, 144, 10]-code), using
- 2 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 0) [i] based on linear OA(8²⁰, 162, F₈, 9) (dual of [162, 142, 10]-code), using
  - trace code [i] based on linear OA(64¹⁰, 81, F₆₄, 9) (dual of [81, 71, 10]-code), using
    - extended algebraic-geometric code AG_e(F,71P) [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]

There is no (12, 21, 10360)-net in base 8, because

1 times m-reduction [i] would yield (12, 20, 10360)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 153366 197441 188431 > 8²⁰ [i]