MinT - Best Known (11, 11+7, s)-Nets in Base 8

Best Known (11, 11+7, s)-Nets in Base 8

(11, 11+7, 195)-Net over F₈ — Constructive and digital

Digital (11, 18, 195)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 4, 65)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁴, 65, F₈, 3, 3) (dual of [(65, 3), 191, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(8⁴, 65, F₈, 2, 3) (dual of [(65, 2), 126, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(8⁴, 65, F₈, 3) (dual of [65, 61, 4]-code or 65-cap in PG(3,8)), using
        ovoid in PG(3,Â 8) [i]
2. digital (7, 14, 130)-net over F₈, using
  - trace code for nets [i] based on digital (0, 7, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
      - Niederreiter sequence [i]

(11, 11+7, 227)-Net over F₈ — Digital

Digital (11, 18, 227)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁸, 227, F₈, 7) (dual of [227, 209, 8]-code), using
- 63 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 14 times 0, 1, 47 times 0) [i] based on linear OA(8¹⁶, 162, F₈, 7) (dual of [162, 146, 8]-code), using
  - trace code [i] based on linear OA(64⁸, 81, F₆₄, 7) (dual of [81, 73, 8]-code), using
    - extended algebraic-geometric code AG_e(F,73P) [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]

(11, 11+7, 258)-Net in Base 8 — Constructive

(11, 18, 258)-net in base 8, using

trace code for nets [i] based on (2, 9, 129)-net in base 64, using
- 5 times m-reduction [i] based on (2, 14, 129)-net in base 64, using
  - base change [i] based on digital (0, 12, 129)-net over F₁₂₈, using
    - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
      - Niederreiter sequence [i]

(11, 11+7, 34023)-Net in Base 8 — Upper bound on s

There is no (11, 18, 34024)-net in base 8, because

1 times m-reduction [i] would yield (11, 17, 34024)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2251 923749 199405 > 8¹⁷ [i]