MinT - Best Known (73−19, 73, s)-Nets in Base 8

Best Known (73−19, 73, s)-Nets in Base 8

(73−19, 73, 514)-Net over F₈ — Constructive and digital

Digital (54, 73, 514)-net over F₈, using

8¹ times duplication [i] based on digital (53, 72, 514)-net over F₈, 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. 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]

(73−19, 73, 674)-Net in Base 8 — Constructive

(54, 73, 674)-net in base 8, using

8¹ times duplication [i] based on (53, 72, 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. (33, 52, 514)-net in base 8, using
    - base change [i] based on digital (20, 39, 514)-net over F₁₆, using
      - 1 times m-reduction [i] based on digital (20, 40, 514)-net over F₁₆, using
        trace code for nets [i] based on digital (0, 20, 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]

(73−19, 73, 5010)-Net over F₈ — Digital

Digital (54, 73, 5010)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁷³, 5010, F₈, 19) (dual of [5010, 4937, 20]-code), using
- 902 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 5 times 0, 1, 16 times 0, 1, 48 times 0, 1, 123 times 0, 1, 264 times 0, 1, 438 times 0) [i] based on linear OA(8⁶⁵, 4100, F₈, 19) (dual of [4100, 4035, 20]-code), using
  - constructionÂ X applied to C_e(18) âŠ‚ C_e(17) [i] based on
    1. linear OA(8⁶⁵, 4096, F₈, 19) (dual of [4096, 4031, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    2. 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]
    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]

(73−19, 73, large)-Net in Base 8 — Upper bound on s

There is no (54, 73, large)-net in base 8, because

17 times m-reduction [i] would yield (54, 56, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]