MinT - Best Known (133−37, 133, s)-Nets in Base 8

Best Known (133−37, 133, s)-Nets in Base 8

(133−37, 133, 1026)-Net over F₈ — Constructive and digital

Digital (96, 133, 1026)-net over F₈, using

3 times m-reduction [i] based on digital (96, 136, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 68, 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]

(133−37, 133, 4451)-Net over F₈ — Digital

Digital (96, 133, 4451)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹³³, 4451, F₈, 37) (dual of [4451, 4318, 38]-code), using
- 347 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 0, 0, 1, 23 times 0, 1, 107 times 0, 1, 211 times 0) [i] based on linear OA(8¹²⁹, 4100, F₈, 37) (dual of [4100, 3971, 38]-code), using
  - constructionÂ X applied to C_e(36) âŠ‚ C_e(35) [i] based on
    1. linear OA(8¹²⁹, 4096, F₈, 37) (dual of [4096, 3967, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    2. linear OA(8¹²⁵, 4096, F₈, 36) (dual of [4096, 3971, 37]-code), using an extension C_e(35) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [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]

(133−37, 133, 4525753)-Net in Base 8 — Upper bound on s

There is no (96, 133, 4525754)-net in base 8, because

1 times m-reduction [i] would yield (96, 132, 4525754)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 161391 186458 771300 117964 732311 623831 970758 722800 224586 976343 856395 144101 794087 780969 356517 426272 076122 943946 673222 011140 > 8¹³² [i]