MinT - Best Known (61, 86, s)-Nets in Base 8

Best Known (61, 86, s)-Nets in Base 8

(61, 86, 416)-Net over F₈ — Constructive and digital

Digital (61, 86, 416)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (18, 30, 208)-net over F₈, using
  - trace code for nets [i] based on digital (3, 15, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]
2. digital (31, 56, 208)-net over F₈, using
  - trace code for nets [i] based on digital (3, 28, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄ (see above)

(61, 86, 576)-Net in Base 8 — Constructive

(61, 86, 576)-net in base 8, using

4 times m-reduction [i] based on (61, 90, 576)-net in base 8, using
- trace code for nets [i] based on (16, 45, 288)-net in base 64, using
  - 4 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
    - base change [i] based on digital (9, 42, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(61, 86, 2916)-Net over F₈ — Digital

Digital (61, 86, 2916)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸⁶, 2916, F₈, 25) (dual of [2916, 2830, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁸⁶, 4101, F₈, 25) (dual of [4101, 4015, 26]-code), using
  - constructionÂ X applied to C_e(24) âŠ‚ C_e(22) [i] based on
    1. linear OA(8⁸⁵, 4096, F₈, 25) (dual of [4096, 4011, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
    2. linear OA(8⁸¹, 4096, F₈, 23) (dual of [4096, 4015, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    3. linear OA(8¹, 5, F₈, 1) (dual of [5, 4, 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]

(61, 86, 1884295)-Net in Base 8 — Upper bound on s

There is no (61, 86, 1884296)-net in base 8, because

1 times m-reduction [i] would yield (61, 85, 1884296)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 57896 056508 576913 491685 492763 977946 842614 403751 339638 640091 860474 931460 583664 > 8⁸⁵ [i]