MinT - Best Known (157−37, 157, s)-Nets in Base 8

Best Known (157−37, 157, s)-Nets in Base 8

(157−37, 157, 1071)-Net over F₈ — Constructive and digital

Digital (120, 157, 1071)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 27, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
      - an explicitly constructive algebraic function field [i]
2. digital (93, 130, 1026)-net over F₈, using
  - trace code for nets [i] based on digital (28, 65, 513)-net over F₆₄, using
    - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(157−37, 157, 1090)-Net in Base 8 — Constructive

(120, 157, 1090)-net in base 8, using

8¹ times duplication [i] based on (119, 156, 1090)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. (30, 48, 514)-net in base 8, using
    - base change [i] based on digital (18, 36, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 18, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
  2. (71, 108, 576)-net in base 8, using
    - trace code for nets [i] based on (17, 54, 288)-net in base 64, using
      - 2 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
        base change [i] based on digital (9, 48, 288)-net over F₁₂₈, using
        net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(157−37, 157, 17724)-Net over F₈ — Digital

Digital (120, 157, 17724)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(157−37, 157, large)-Net in Base 8 — Upper bound on s

There is no (120, 157, large)-net in base 8, because

35 times m-reduction [i] would yield (120, 122, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]