MinT - Best Known (81−22, 81, s)-Nets in Base 8

Best Known (81−22, 81, s)-Nets in Base 8

(81−22, 81, 484)-Net over F₈ — Constructive and digital

Digital (59, 81, 484)-net over F₈, using

1 times m-reduction [i] based on digital (59, 82, 484)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 22, 130)-net over F₈, using
    - trace code for nets [i] based on digital (0, 11, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [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]

(81−22, 81, 579)-Net in Base 8 — Constructive

(59, 81, 579)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (10, 21, 65)-net over F₈, using
  - base reduction for projective spaces (embedding PG(10,64) in PG(20,8)) for nets [i] based on digital (0, 11, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
      - Niederreiter sequence [i]
2. (38, 60, 514)-net in base 8, using
  - base change [i] based on digital (23, 45, 514)-net over F₁₆, using
    - 1 times m-reduction [i] based on digital (23, 46, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 23, 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]

(81−22, 81, 4203)-Net over F₈ — Digital

Digital (59, 81, 4203)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸¹, 4203, F₈, 22) (dual of [4203, 4122, 23]-code), using
- 99 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 5 times 0, 1, 23 times 0, 1, 68 times 0) [i] based on linear OA(8⁷⁷, 4100, F₈, 22) (dual of [4100, 4023, 23]-code), using
  - constructionÂ X applied to C_e(21) âŠ‚ C_e(20) [i] based on
    1. 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]
    2. linear OA(8⁷³, 4096, F₈, 21) (dual of [4096, 4023, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [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]

(81−22, 81, 3132863)-Net in Base 8 — Upper bound on s

There is no (59, 81, 3132864)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 14 134801 286621 422846 602548 090603 969913 655917 385280 966415 567151 839912 782713 > 8⁸¹ [i]