MinT - Best Known (94, 94+37, s)-Nets in Base 8

Best Known (94, 94+37, s)-Nets in Base 8

(94, 94+37, 1026)-Net over F₈ — Constructive and digital

Digital (94, 131, 1026)-net over F₈, using

1 times m-reduction [i] based on digital (94, 132, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 66, 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]

(94, 94+37, 4131)-Net over F₈ — Digital

Digital (94, 131, 4131)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹³¹, 4131, F₈, 37) (dual of [4131, 4000, 38]-code), using
- 24 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 23 times 0) [i] based on linear OA(8¹³⁰, 4106, F₈, 37) (dual of [4106, 3976, 38]-code), using
  - constructionÂ X applied to C([0,18]) âŠ‚ C([0,17]) [i] based on
    1. linear OA(8¹²⁹, 4097, F₈, 37) (dual of [4097, 3968, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
    2. linear OA(8¹²¹, 4097, F₈, 35) (dual of [4097, 3976, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,17], and minimum distance dÂ â‰¥ |{−17,−16,…,17}|+1Â =Â 36 (BCH-bound) [i]
    3. linear OA(8¹, 9, F₈, 1) (dual of [9, 8, 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]

(94, 94+37, 3592090)-Net in Base 8 — Upper bound on s

There is no (94, 131, 3592091)-net in base 8, because

1 times m-reduction [i] would yield (94, 130, 3592091)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2521 736704 000728 056078 490839 130762 878487 051189 489301 320344 350245 041361 917513 010288 289190 304245 994220 910354 982402 481835 > 8¹³⁰ [i]