MinT - Best Known (28−9, 28, s)-Nets in Base 8

Best Known (28−9, 28, s)-Nets in Base 8

(28−9, 28, 290)-Net over F₈ — Constructive and digital

Digital (19, 28, 290)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 8, 130)-net over F₈, using
  - trace code for nets [i] based on digital (0, 4, 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 (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]

(28−9, 28, 523)-Net in Base 8 — Constructive

(19, 28, 523)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (0, 4, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. (15, 24, 514)-net in base 8, using
  - base change [i] based on digital (9, 18, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 9, 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]

(28−9, 28, 799)-Net over F₈ — Digital

Digital (19, 28, 799)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8²⁸, 799, F₈, 9) (dual of [799, 771, 10]-code), using
- 281 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 12 times 0, 1, 35 times 0, 1, 80 times 0, 1, 146 times 0) [i] based on linear OA(8²², 512, F₈, 9) (dual of [512, 490, 10]-code), using
  - an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(28−9, 28, 394284)-Net in Base 8 — Upper bound on s

There is no (19, 28, 394285)-net in base 8, because

1 times m-reduction [i] would yield (19, 27, 394285)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2 417859 528984 049639 689831 > 8²⁷ [i]