MinT - Best Known (80, 80+31, s)-Nets in Base 8

Best Known (80, 80+31, s)-Nets in Base 8

(80, 80+31, 514)-Net over F₈ — Constructive and digital

Digital (80, 111, 514)-net over F₈, using

1 times m-reduction [i] based on digital (80, 112, 514)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (18, 34, 160)-net over F₈, using
    - trace code for nets [i] based on digital (1, 17, 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 (46, 78, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 39, 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]

(80, 80+31, 593)-Net in Base 8 — Constructive

(80, 111, 593)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (2, 17, 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. (63, 94, 576)-net in base 8, using
  - trace code for nets [i] based on (16, 47, 288)-net in base 64, using
    - 2 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]

(80, 80+31, 4116)-Net over F₈ — Digital

Digital (80, 111, 4116)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹¹, 4116, F₈, 31) (dual of [4116, 4005, 32]-code), using
- 10 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 9 times 0) [i] based on linear OA(8¹¹⁰, 4105, F₈, 31) (dual of [4105, 3995, 32]-code), using
  - constructionÂ X applied to C_e(30) âŠ‚ C_e(28) [i] based on
    1. linear OA(8¹⁰⁹, 4096, F₈, 31) (dual of [4096, 3987, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
    2. linear OA(8¹⁰¹, 4096, F₈, 29) (dual of [4096, 3995, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    3. linear OA(8¹, 9, F₈, 1) (dual of [9, 8, 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]

(80, 80+31, 3848818)-Net in Base 8 — Upper bound on s

There is no (80, 111, 3848819)-net in base 8, because

1 times m-reduction [i] would yield (80, 110, 3848819)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2187 251794 286805 798153 940134 921609 517528 160635 131772 823176 509532 682076 119521 204402 400852 365747 199120 > 8¹¹⁰ [i]