MinT - Best Known (126−35, 126, s)-Nets in Base 8

Best Known (126−35, 126, s)-Nets in Base 8

(126−35, 126, 1026)-Net over F₈ — Constructive and digital

Digital (91, 126, 1026)-net over F₈, using

trace code for nets [i] based on digital (28, 63, 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]

(126−35, 126, 4366)-Net over F₈ — Digital

Digital (91, 126, 4366)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹²⁶, 4366, F₈, 35) (dual of [4366, 4240, 36]-code), using
- 261 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 5 times 0, 1, 26 times 0, 1, 74 times 0, 1, 152 times 0) [i] based on linear OA(8¹²¹, 4100, F₈, 35) (dual of [4100, 3979, 36]-code), using
  - constructionÂ X applied to C_e(34) âŠ‚ C_e(33) [i] based on
    1. linear OA(8¹²¹, 4096, F₈, 35) (dual of [4096, 3975, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
    2. linear OA(8¹¹⁷, 4096, F₈, 34) (dual of [4096, 3979, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [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]

(126−35, 126, 4479336)-Net in Base 8 — Upper bound on s

There is no (91, 126, 4479337)-net in base 8, because

1 times m-reduction [i] would yield (91, 125, 4479337)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 76957 135128 734330 661008 059105 869779 170333 037325 443776 342167 903726 149358 308964 219918 405846 861621 842776 744315 356576 > 8¹²⁵ [i]