MinT - Best Known (86−23, 86, s)-Nets in Base 8

Best Known (86−23, 86, s)-Nets in Base 8

(86−23, 86, 514)-Net over F₈ — Constructive and digital

Digital (63, 86, 514)-net over F₈, using

8² times duplication [i] based on digital (61, 84, 514)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. 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]
  2. digital (37, 60, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 30, 177)-net over F₆₄, using
      - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(86−23, 86, 674)-Net in Base 8 — Constructive

(63, 86, 674)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. 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]
2. (39, 62, 514)-net in base 8, using
  - trace code for nets [i] based on (8, 31, 257)-net in base 64, using
    - 1 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
      - base change [i] based on digital (0, 24, 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]

(86−23, 86, 4489)-Net over F₈ — Digital

Digital (63, 86, 4489)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸⁶, 4489, F₈, 23) (dual of [4489, 4403, 24]-code), using
- 384 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 6 times 0, 1, 34 times 0, 1, 108 times 0, 1, 232 times 0) [i] based on linear OA(8⁸¹, 4100, F₈, 23) (dual of [4100, 4019, 24]-code), using
  - constructionÂ X applied to C_e(22) âŠ‚ C_e(21) [i] based on
    1. 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]
    2. linear OA(8⁷⁷, 4096, F₈, 22) (dual of [4096, 4019, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    3. linear OA(8⁰, 4, F₈, 0) (dual of [4, 4, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(86−23, 86, 6673264)-Net in Base 8 — Upper bound on s

There is no (63, 86, 6673265)-net in base 8, because

1 times m-reduction [i] would yield (63, 85, 6673265)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 57896 126839 903138 910229 054264 650050 501174 757186 839017 405555 307918 291126 973072 > 8⁸⁵ [i]