MinT - Best Known (109, 109+44, s)-Nets in Base 8

Best Known (109, 109+44, s)-Nets in Base 8

(109, 109+44, 1026)-Net over F₈ — Constructive and digital

Digital (109, 153, 1026)-net over F₈, using

9 times m-reduction [i] based on digital (109, 162, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 81, 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]

(109, 109+44, 4100)-Net over F₈ — Digital

Digital (109, 153, 4100)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁵³, 4100, F₈, 44) (dual of [4100, 3947, 45]-code), using
- constructionÂ X applied to C_e(43) âŠ‚ C_e(42) [i] based on
  1. linear OA(8¹⁵³, 4096, F₈, 44) (dual of [4096, 3943, 45]-code), using an extension C_e(43) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,43], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 44 [i]
  2. linear OA(8¹⁴⁹, 4096, F₈, 43) (dual of [4096, 3947, 44]-code), using an extension C_e(42) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,42], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 43 [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]

(109, 109+44, 2467930)-Net in Base 8 — Upper bound on s

There is no (109, 153, 2467931)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 488570 461326 913525 744111 503366 339119 039694 711397 331831 160651 789686 287503 321344 990779 183809 918203 310744 370900 794714 366132 506820 431708 896478 > 8¹⁵³ [i]