MinT - Best Known (111−28, 111, s)-Nets in Base 8

Best Known (111−28, 111, s)-Nets in Base 8

(111−28, 111, 610)-Net over F₈ — Constructive and digital

Digital (83, 111, 610)-net over F₈, using

8¹ times duplication [i] based on digital (82, 110, 610)-net over F₈, using
- t-expansion [i] based on digital (81, 110, 610)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (24, 38, 256)-net over F₈, using
      - trace code for nets [i] based on digital (5, 19, 128)-net over F₆₄, using
        net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
        a function field by SÃ©mirat [i]
    2. digital (43, 72, 354)-net over F₈, using
      - trace code for nets [i] based on digital (7, 36, 177)-net over F₆₄, using
        net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(111−28, 111, 772)-Net in Base 8 — Constructive

(83, 111, 772)-net in base 8, using

1 times m-reduction [i] based on (83, 112, 772)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. (20, 34, 258)-net in base 8, using
    - trace code for nets [i] based on (3, 17, 129)-net in base 64, using
      - 4 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
        base change [i] based on digital (0, 18, 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]
  2. (49, 78, 514)-net in base 8, using
    - trace code for nets [i] based on (10, 39, 257)-net in base 64, using
      - 1 times m-reduction [i] based on (10, 40, 257)-net in base 64, using
        base change [i] based on digital (0, 30, 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]

(111−28, 111, 8198)-Net over F₈ — Digital

Digital (83, 111, 8198)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹¹, 8198, F₈, 28) (dual of [8198, 8087, 29]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(8¹¹⁰, 8196, F₈, 28) (dual of [8196, 8086, 29]-code), using
    - trace code [i] based on linear OA(64⁵⁵, 4098, F₆₄, 28) (dual of [4098, 4043, 29]-code), using
      - constructionÂ X applied to C_e(27) âŠ‚ C_e(26) [i] based on
        linear OA(64⁵⁵, 4096, F₆₄, 28) (dual of [4096, 4041, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
        linear OA(64⁵³, 4096, F₆₄, 27) (dual of [4096, 4043, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
        linear OA(64⁰, 2, F₆₄, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
  2. linear OA(8¹¹⁰, 8197, F₈, 27) (dual of [8197, 8087, 28]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 8¹¹⁰Â > V_b^sâˆ’1(kâˆ’1)Â = 12 692260 739715 569336 270810 944542 550803 151319 280869 369688 711687 580374 063558 045795 800637 909133 491200 [i]
  3. linear OA(8⁰, 1, F₈, 0) (dual of [1, 1, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]

(111−28, 111, large)-Net in Base 8 — Upper bound on s

There is no (83, 111, large)-net in base 8, because

26 times m-reduction [i] would yield (83, 85, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (111−28, 111, s)-Nets in Base 8

(111−28, 111, 610)-Net over F8 — Constructive and digital

(111−28, 111, 772)-Net in Base 8 — Constructive

(111−28, 111, 8198)-Net over F8 — Digital

(111−28, 111, large)-Net in Base 8 — Upper bound on s

(111−28, 111, 610)-Net over F₈ — Constructive and digital

(111−28, 111, 8198)-Net over F₈ — Digital