MinT - Best Known (80, 80+30, s)-Nets in Base 8

Best Known (80, 80+30, s)-Nets in Base 8

(80, 80+30, 562)-Net over F₈ — Constructive and digital

Digital (80, 110, 562)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (21, 36, 208)-net over F₈, using
  - trace code for nets [i] based on digital (3, 18, 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 (44, 74, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 37, 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+30, 644)-Net in Base 8 — Constructive

(80, 110, 644)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (15, 30, 130)-net over F₈, using
  - trace code for nets [i] based on digital (0, 15, 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. (50, 80, 514)-net in base 8, using
  - base change [i] based on digital (30, 60, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 30, 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]

(80, 80+30, 4491)-Net over F₈ — Digital

Digital (80, 110, 4491)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹⁰, 4491, F₈, 30) (dual of [4491, 4381, 31]-code), using
- 386 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 6 times 0, 1, 38 times 0, 1, 117 times 0, 1, 221 times 0) [i] based on linear OA(8¹⁰⁵, 4100, F₈, 30) (dual of [4100, 3995, 31]-code), using
  - constructionÂ X applied to C_e(29) âŠ‚ C_e(28) [i] based on
    1. linear OA(8¹⁰⁵, 4096, F₈, 30) (dual of [4096, 3991, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [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(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]

(80, 80+30, 3848818)-Net in Base 8 — Upper bound on s

There is no (80, 110, 3848819)-net in base 8, because

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]