MinT - Best Known (87, 87+34, s)-Nets in Base 8

Best Known (87, 87+34, s)-Nets in Base 8

(87, 87+34, 514)-Net over F₈ — Constructive and digital

Digital (87, 121, 514)-net over F₈, using

8¹ times duplication [i] based on digital (86, 120, 514)-net over F₈, using
- t-expansion [i] based on digital (85, 120, 514)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (19, 36, 160)-net over F₈, using
      - trace code for nets [i] based on digital (1, 18, 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 (49, 84, 354)-net over F₈, using
      - trace code for nets [i] based on digital (7, 42, 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]

(87, 87+34, 593)-Net in Base 8 — Constructive

(87, 121, 593)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (2, 19, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. (68, 102, 576)-net in base 8, using
  - trace code for nets [i] based on (17, 51, 288)-net in base 64, using
    - 5 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
      - base change [i] based on digital (9, 48, 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]

(87, 87+34, 4160)-Net over F₈ — Digital

Digital (87, 121, 4160)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹²¹, 4160, F₈, 34) (dual of [4160, 4039, 35]-code), using
- 56 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 13 times 0, 1, 37 times 0) [i] based on linear OA(8¹¹⁷, 4100, F₈, 34) (dual of [4100, 3983, 35]-code), using
  - constructionÂ X applied to C_e(33) âŠ‚ C_e(32) [i] based on
    1. linear OA(8¹¹⁷, 4096, F₈, 34) (dual of [4096, 3979, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    2. linear OA(8¹¹³, 4096, F₈, 33) (dual of [4096, 3983, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [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]

(87, 87+34, 2746131)-Net in Base 8 — Upper bound on s

There is no (87, 121, 2746132)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 18 788415 401395 886071 589144 947417 091922 115226 336057 970776 697365 836657 274954 986138 297635 677978 251981 102020 996741 > 8¹²¹ [i]