MinT - Best Known (149−42, 149, s)-Nets in Base 8

Best Known (149−42, 149, s)-Nets in Base 8

(149−42, 149, 1026)-Net over F₈ — Constructive and digital

Digital (107, 149, 1026)-net over F₈, using

9 times m-reduction [i] based on digital (107, 158, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 79, 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]

(149−42, 149, 4438)-Net over F₈ — Digital

Digital (107, 149, 4438)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁴⁹, 4438, F₈, 42) (dual of [4438, 4289, 43]-code), using
- 334 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 0, 0, 0, 1, 27 times 0, 1, 108 times 0, 1, 192 times 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]

(149−42, 149, 3169795)-Net in Base 8 — Upper bound on s

There is no (107, 149, 3169796)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 363 420905 374023 500615 399529 329093 620455 972575 026703 111288 998116 856239 869416 103210 401185 819811 118131 029915 654776 526326 114279 685616 695168 > 8¹⁴⁹ [i]