MinT - Best Known (146−42, 146, s)-Nets in Base 8

Best Known (146−42, 146, s)-Nets in Base 8

(146−42, 146, 1026)-Net over F₈ — Constructive and digital

Digital (104, 146, 1026)-net over F₈, using

6 times m-reduction [i] based on digital (104, 152, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 76, 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]

(146−42, 146, 4105)-Net over F₈ — Digital

Digital (104, 146, 4105)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁴⁶, 4105, F₈, 42) (dual of [4105, 3959, 43]-code), using
- 4 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 0, 0, 0) [i] based on linear OA(8¹⁴⁵, 4100, F₈, 42) (dual of [4100, 3955, 43]-code), using
  - constructionÂ X applied to C_e(41) âŠ‚ C_e(40) [i] based on
    1. linear OA(8¹⁴⁵, 4096, F₈, 42) (dual of [4096, 3951, 43]-code), using an extension C_e(41) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,41], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 42 [i]
    2. linear OA(8¹⁴¹, 4096, F₈, 41) (dual of [4096, 3955, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [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]

(146−42, 146, 2355145)-Net in Base 8 — Upper bound on s

There is no (104, 146, 2355146)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 709806 228991 357384 474389 734222 828894 449353 488654 665350 645495 498352 182437 247336 533702 277065 458089 936450 059722 641225 727069 608848 461448 > 8¹⁴⁶ [i]