MinT - Best Known (22−9, 22, s)-Nets in Base 8

Best Known (22−9, 22, s)-Nets in Base 8

(22−9, 22, 160)-Net over F₈ — Constructive and digital

Digital (13, 22, 160)-net over F₈, using

2 times m-reduction [i] based on digital (13, 24, 160)-net over F₈, using
- trace code for nets [i] based on digital (1, 12, 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]

(22−9, 22, 256)-Net over F₈ — Digital

Digital (13, 22, 256)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(8²², 256, F₈, 2, 9) (dual of [(256, 2), 490, 10]-NRT-code), using
- OOA 2-folding [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]

(22−9, 22, 258)-Net in Base 8 — Constructive

(13, 22, 258)-net in base 8, using

trace code for nets [i] based on (2, 11, 129)-net in base 64, using
- 3 times m-reduction [i] based on (2, 14, 129)-net in base 64, using
  - base change [i] based on digital (0, 12, 129)-net over F₁₂₈, using
    - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
      - Niederreiter sequence [i]

(22−9, 22, 17423)-Net in Base 8 — Upper bound on s

There is no (13, 22, 17424)-net in base 8, because

1 times m-reduction [i] would yield (13, 21, 17424)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 9 225265 387800 002773 > 8²¹ [i]