MinT - Best Known (26−8, 26, s)-Nets in Base 8

Best Known (26−8, 26, s)-Nets in Base 8

(26−8, 26, 290)-Net over F₈ — Constructive and digital

Digital (18, 26, 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 (10, 18, 160)-net over F₈, using
  - trace code for nets [i] based on digital (1, 9, 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]

(26−8, 26, 523)-Net in Base 8 — Constructive

(18, 26, 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. (14, 22, 514)-net in base 8, using
  - trace code for nets [i] based on (3, 11, 257)-net in base 64, using
    - 1 times m-reduction [i] based on (3, 12, 257)-net in base 64, using
      - base change [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]

(26−8, 26, 1096)-Net over F₈ — Digital

Digital (18, 26, 1096)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8²⁶, 1096, F₈, 8) (dual of [1096, 1070, 9]-code), using
- 580 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 0, 0, 1, 20 times 0, 1, 90 times 0, 1, 188 times 0, 1, 275 times 0) [i] based on linear OA(8²¹, 511, F₈, 8) (dual of [511, 490, 9]-code), using
  - 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]

(26−8, 26, 234442)-Net in Base 8 — Upper bound on s

There is no (18, 26, 234443)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 302235 411710 559255 721794 > 8²⁶ [i]