MinT - Best Known (77, 106, s)-Nets in Base 8

Best Known (77, 106, s)-Nets in Base 8

(77, 106, 562)-Net over F₈ — Constructive and digital

Digital (77, 106, 562)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (20, 34, 208)-net over F₈, using
  - trace code for nets [i] based on digital (3, 17, 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 (43, 72, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 36, 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]

(77, 106, 644)-Net in Base 8 — Constructive

(77, 106, 644)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (14, 28, 130)-net over F₈, using
  - trace code for nets [i] based on digital (0, 14, 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. (49, 78, 514)-net in base 8, using
  - trace code for nets [i] based on (10, 39, 257)-net in base 64, using
    - 1 times m-reduction [i] based on (10, 40, 257)-net in base 64, using
      - base change [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]

(77, 106, 4351)-Net over F₈ — Digital

Digital (77, 106, 4351)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁰⁶, 4351, F₈, 29) (dual of [4351, 4245, 30]-code), using
- 246 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 5 times 0, 1, 23 times 0, 1, 66 times 0, 1, 148 times 0) [i] based on linear OA(8¹⁰¹, 4100, F₈, 29) (dual of [4100, 3999, 30]-code), using
  - constructionÂ X applied to C_e(28) âŠ‚ C_e(27) [i] based on
    1. 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]
    2. linear OA(8⁹⁷, 4096, F₈, 28) (dual of [4096, 3999, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [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]

(77, 106, 5123112)-Net in Base 8 — Upper bound on s

There is no (77, 106, 5123113)-net in base 8, because

1 times m-reduction [i] would yield (77, 105, 5123113)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 66749 710790 953332 599893 687196 300220 024087 361288 361983 605624 024275 362676 092232 525765 860394 250928 > 8¹⁰⁵ [i]