MinT - Best Known (31−13, 31, s)-Nets in Base 9

Best Known (31−13, 31, s)-Nets in Base 9

(31−13, 31, 232)-Net over F₉ — Constructive and digital

Digital (18, 31, 232)-net over F₉, using

1 times m-reduction [i] based on digital (18, 32, 232)-net over F₉, using
- trace code for nets [i] based on digital (2, 16, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(31−13, 31, 238)-Net over F₉ — Digital

Digital (18, 31, 238)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9³¹, 238, F₉, 13) (dual of [238, 207, 14]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(9³⁰, 236, F₉, 13) (dual of [236, 206, 14]-code), using
    - trace code [i] based on linear OA(81¹⁵, 118, F₈₁, 13) (dual of [118, 103, 14]-code), using
      - extended algebraic-geometric code AG_e(F,104P) [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 118, using
        sharp bound on the number of rational points on curves with genus 2 [i]
  2. linear OA(9³⁰, 237, F₉, 12) (dual of [237, 207, 13]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 9³⁰Â > V_b^sâˆ’1(kâˆ’1)Â = 21615 586080 098392 378378 716129 [i]
  3. linear OA(9⁰, 1, F₉, 0) (dual of [1, 1, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(9⁰, s, F₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]

(31−13, 31, 22094)-Net in Base 9 — Upper bound on s

There is no (18, 31, 22095)-net in base 9, because

1 times m-reduction [i] would yield (18, 30, 22095)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 42398 910938 576726 207377 455313 > 9³⁰ [i]