MinT - Best Known (58, 80, s)-Nets in Base 8

Best Known (58, 80, s)-Nets in Base 8

(58, 80, 484)-Net over F₈ — Constructive and digital

Digital (58, 80, 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 (36, 58, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 29, 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]

(58, 80, 576)-Net in Base 8 — Constructive

(58, 80, 576)-net in base 8, using

t-expansion [i] based on (57, 80, 576)-net in base 8, using
- 4 times m-reduction [i] based on (57, 84, 576)-net in base 8, using
  - trace code for nets [i] based on (15, 42, 288)-net in base 64, using
    - base change [i] based on digital (9, 36, 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]

(58, 80, 4133)-Net over F₈ — Digital

Digital (58, 80, 4133)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸⁰, 4133, F₈, 22) (dual of [4133, 4053, 23]-code), using
- 30 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 5 times 0, 1, 23 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]

(58, 80, 2593236)-Net in Base 8 — Upper bound on s

There is no (58, 80, 2593237)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 766847 766093 433936 325468 865129 578325 445595 112075 490885 729572 803985 007080 > 8⁸⁰ [i]