MinT - Best Known (33, 53, s)-Nets in Base 8

Best Known (33, 53, s)-Nets in Base 8

(33, 53, 256)-Net over F₈ — Constructive and digital

Digital (33, 53, 256)-net over F₈, using

3 times m-reduction [i] based on digital (33, 56, 256)-net over F₈, using
- trace code for nets [i] based on digital (5, 28, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(33, 53, 300)-Net in Base 8 — Constructive

(33, 53, 300)-net in base 8, using

3 times m-reduction [i] based on (33, 56, 300)-net in base 8, using
- trace code for nets [i] based on (5, 28, 150)-net in base 64, using
  - base change [i] based on digital (1, 24, 150)-net over F₁₂₈, using
    - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]

(33, 53, 429)-Net over F₈ — Digital

Digital (33, 53, 429)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁵³, 429, F₈, 20) (dual of [429, 376, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁵³, 519, F₈, 20) (dual of [519, 466, 21]-code), using
  - constructionÂ X applied to C_e(19) âŠ‚ C_e(17) [i] based on
    1. linear OA(8⁵², 512, F₈, 20) (dual of [512, 460, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    2. linear OA(8⁴⁶, 512, F₈, 18) (dual of [512, 466, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
    3. linear OA(8¹, 7, F₈, 1) (dual of [7, 6, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹, s, F₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(33, 53, 39553)-Net in Base 8 — Upper bound on s

There is no (33, 53, 39554)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 730755 887711 979530 399861 063923 416147 180892 102464 > 8⁵³ [i]