MinT - Best Known (139−39, 139, s)-Nets in Base 8

Best Known (139−39, 139, s)-Nets in Base 8

(139−39, 139, 1026)-Net over F₈ — Constructive and digital

Digital (100, 139, 1026)-net over F₈, using

5 times m-reduction [i] based on digital (100, 144, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 72, 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]

(139−39, 139, 4347)-Net over F₈ — Digital

Digital (100, 139, 4347)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹³⁹, 4347, F₈, 39) (dual of [4347, 4208, 40]-code), using
- 245 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 69 times 0, 1, 174 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]

(139−39, 139, 4105834)-Net in Base 8 — Upper bound on s

There is no (100, 139, 4105835)-net in base 8, because

1 times m-reduction [i] would yield (100, 138, 4105835)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 42307 760709 665985 914968 935039 735209 673532 114411 997759 309664 120148 473224 656467 374372 808929 096099 562724 700116 860201 188838 499296 > 8¹³⁸ [i]