MinT - Best Known (143−41, 143, s)-Nets in Base 8

Best Known (143−41, 143, s)-Nets in Base 8

(143−41, 143, 1026)-Net over F₈ — Constructive and digital

Digital (102, 143, 1026)-net over F₈, using

5 times m-reduction [i] based on digital (102, 148, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 74, 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]

(143−41, 143, 4107)-Net over F₈ — Digital

Digital (102, 143, 4107)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁴³, 4107, F₈, 41) (dual of [4107, 3964, 42]-code), using
- 5 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 4 times 0) [i] based on linear OA(8¹⁴², 4101, F₈, 41) (dual of [4101, 3959, 42]-code), using
  - constructionÂ X applied to C_e(40) âŠ‚ C_e(38) [i] based on
    1. 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]
    2. 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]
    3. linear OA(8¹, 5, F₈, 1) (dual of [5, 4, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹, s, F₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(143−41, 143, 3062988)-Net in Base 8 — Upper bound on s

There is no (102, 143, 3062989)-net in base 8, because

1 times m-reduction [i] would yield (102, 142, 3062989)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 173 292580 163922 324032 278746 096201 961119 956465 544165 141084 584077 309368 395656 388313 152028 057500 752737 069926 353871 743645 273100 917684 > 8¹⁴² [i]