MinT - Best Known (133−38, 133, s)-Nets in Base 8

Best Known (133−38, 133, s)-Nets in Base 8

(133−38, 133, 1026)-Net over F₈ — Constructive and digital

Digital (95, 133, 1026)-net over F₈, using

1 times m-reduction [i] based on digital (95, 134, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 67, 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]

(133−38, 133, 4100)-Net over F₈ — Digital

Digital (95, 133, 4100)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹³³, 4100, F₈, 38) (dual of [4100, 3967, 39]-code), using
- constructionÂ X applied to C_e(37) âŠ‚ C_e(36) [i] based on
  1. 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]
  2. 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]
  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]

(133−38, 133, 2375446)-Net in Base 8 — Upper bound on s

There is no (95, 133, 2375447)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 291131 233031 099946 274303 040104 379257 958928 798447 619375 633956 473560 170949 389646 007368 937301 678166 525678 801372 696332 695012 > 8¹³³ [i]