MinT - Best Known (27−9, 27, s)-Nets in Base 8

Best Known (27−9, 27, s)-Nets in Base 8

(27−9, 27, 260)-Net over F₈ — Constructive and digital

Digital (18, 27, 260)-net over F₈, using

8¹ times duplication [i] based on digital (17, 26, 260)-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 (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 (see above)

(27−9, 27, 514)-Net in Base 8 — Constructive

(18, 27, 514)-net in base 8, using

1 times m-reduction [i] based on (18, 28, 514)-net in base 8, using
- base change [i] based on digital (11, 21, 514)-net over F₁₆, using
  - 1 times m-reduction [i] based on digital (11, 22, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 11, 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]

(27−9, 27, 651)-Net over F₈ — Digital

Digital (18, 27, 651)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8²⁷, 651, F₈, 9) (dual of [651, 624, 10]-code), using
- 134 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) [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]

(27−9, 27, 234442)-Net in Base 8 — Upper bound on s

There is no (18, 27, 234443)-net in base 8, because

1 times m-reduction [i] would yield (18, 26, 234443)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 302235 411710 559255 721794 > 8²⁶ [i]