MinT - Best Known (69−18, 69, s)-Nets in Base 8

Best Known (69−18, 69, s)-Nets in Base 8

(69−18, 69, 484)-Net over F₈ — Constructive and digital

Digital (51, 69, 484)-net over F₈, using

1 times m-reduction [i] based on digital (51, 70, 484)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 18, 130)-net over F₈, using
    - trace code for nets [i] based on digital (0, 9, 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]
  2. digital (33, 52, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 26, 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]

(69−18, 69, 674)-Net in Base 8 — Constructive

(51, 69, 674)-net in base 8, using

8¹ times duplication [i] based on (50, 68, 674)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 20, 160)-net over F₈, using
    - trace code for nets [i] based on digital (1, 10, 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]
  2. (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]

(69−18, 69, 4834)-Net over F₈ — Digital

Digital (51, 69, 4834)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁶⁹, 4834, F₈, 18) (dual of [4834, 4765, 19]-code), using
- 726 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 0, 0, 0, 1, 10 times 0, 1, 32 times 0, 1, 86 times 0, 1, 204 times 0, 1, 384 times 0) [i] based on linear OA(8⁶¹, 4100, F₈, 18) (dual of [4100, 4039, 19]-code), using
  - constructionÂ X applied to C_e(17) âŠ‚ C_e(16) [i] based on
    1. linear OA(8⁶¹, 4096, F₈, 18) (dual of [4096, 4035, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
    2. linear OA(8⁵⁷, 4096, F₈, 17) (dual of [4096, 4039, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    3. linear OA(8⁰, 4, F₈, 0) (dual of [4, 4, 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]

(69−18, 69, 4969845)-Net in Base 8 — Upper bound on s

There is no (51, 69, 4969846)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 205 688432 212709 601950 846392 383460 811083 416117 346955 972800 126026 > 8⁶⁹ [i]