MinT - Best Known (140−39, 140, s)-Nets in Base 8

Best Known (140−39, 140, s)-Nets in Base 8

(140−39, 140, 1026)-Net over F₈ — Constructive and digital

Digital (101, 140, 1026)-net over F₈, using

6 times m-reduction [i] based on digital (101, 146, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 73, 513)-net over F₆₄, using
  - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
      - the Hermitian function field over F₆₄ [i]

(140−39, 140, 4580)-Net over F₈ — Digital

Digital (101, 140, 4580)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁴⁰, 4580, F₈, 39) (dual of [4580, 4440, 40]-code), using
- 477 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 69 times 0, 1, 174 times 0, 1, 231 times 0) [i] based on linear OA(8¹³⁷, 4100, F₈, 39) (dual of [4100, 3963, 40]-code), using
  - constructionÂ X applied to C_e(38) âŠ‚ C_e(37) [i] based on
    1. linear OA(8¹³⁷, 4096, F₈, 39) (dual of [4096, 3959, 40]-code), using an extension C_e(38) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]
    2. linear OA(8¹³³, 4096, F₈, 38) (dual of [4096, 3963, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [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]

(140−39, 140, 4580707)-Net in Base 8 — Upper bound on s

There is no (101, 140, 4580708)-net in base 8, because

1 times m-reduction [i] would yield (101, 139, 4580708)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 338461 128766 142879 296112 505759 780108 697719 202460 168939 663079 604192 230661 812896 222049 381631 055720 182168 100035 536873 592229 892414 > 8¹³⁹ [i]