MinT - Best Known (114−32, 114, s)-Nets in Base 8

Best Known (114−32, 114, s)-Nets in Base 8

(114−32, 114, 514)-Net over F₈ — Constructive and digital

Digital (82, 114, 514)-net over F₈, using

t-expansion [i] based on digital (81, 114, 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 (47, 80, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 40, 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]

(114−32, 114, 593)-Net in Base 8 — Constructive

(82, 114, 593)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (2, 18, 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. (64, 96, 576)-net in base 8, using
  - trace code for nets [i] based on (16, 48, 288)-net in base 64, using
    - 1 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]

(114−32, 114, 4106)-Net over F₈ — Digital

Digital (82, 114, 4106)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹⁴, 4106, F₈, 32) (dual of [4106, 3992, 33]-code), using
- 5 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 4 times 0) [i] based on linear OA(8¹¹³, 4100, F₈, 32) (dual of [4100, 3987, 33]-code), using
  - 1 times truncation [i] based on linear OA(8¹¹⁴, 4101, F₈, 33) (dual of [4101, 3987, 34]-code), using
    - constructionÂ X applied to C_e(32) âŠ‚ C_e(30) [i] based on
      1. linear OA(8¹¹³, 4096, F₈, 33) (dual of [4096, 3983, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]
      2. 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]
      3. linear OA(8¹, 5, F₈, 1) (dual of [5, 4, 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]

(114−32, 114, 2642136)-Net in Base 8 — Upper bound on s

There is no (82, 114, 2642137)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 8 959011 799162 336037 184422 540350 454326 141997 463662 261227 117094 060765 280417 772998 417096 587448 391278 118715 > 8¹¹⁴ [i]