MinT - Best Known (142−41, 142, s)-Nets in Base 8

Best Known (142−41, 142, s)-Nets in Base 8

(142−41, 142, 1026)-Net over F₈ — Constructive and digital

Digital (101, 142, 1026)-net over F₈, using

4 times m-reduction [i] based on digital (101, 146, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 73, 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]

(142−41, 142, 4025)-Net over F₈ — Digital

Digital (101, 142, 4025)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁴², 4025, F₈, 41) (dual of [4025, 3883, 42]-code), using
- discarding factors / shortening the dual code 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]

(142−41, 142, 2760518)-Net in Base 8 — Upper bound on s

There is no (101, 142, 2760519)-net in base 8, because

1 times m-reduction [i] would yield (101, 141, 2760519)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 21 661567 347355 921490 624841 364925 835338 567102 999434 831458 718289 322142 079268 503155 618299 871033 725034 268796 468278 006481 349148 809988 > 8¹⁴¹ [i]