MinT - Best Known (154−37, 154, s)-Nets in Base 3

Best Known (154−37, 154, s)-Nets in Base 3

(154−37, 154, 400)-Net over F₃ — Constructive and digital

Digital (117, 154, 400)-net over F₃, using

3² times duplication [i] based on digital (115, 152, 400)-net over F₃, using
- trace code for nets [i] based on digital (1, 38, 100)-net over F₈₁, using
  - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
      - a function field by SÃ©mirat [i]

(154−37, 154, 811)-Net over F₃ — Digital

Digital (117, 154, 811)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁵⁴, 811, F₃, 37) (dual of [811, 657, 38]-code), using
- 64 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 10 times 0, 1, 13 times 0, 1, 16 times 0) [i] based on linear OA(3¹⁴³, 736, F₃, 37) (dual of [736, 593, 38]-code), using
  - constructionÂ X applied to C_e(36) âŠ‚ C_e(34) [i] based on
    1. linear OA(3¹⁴², 729, F₃, 37) (dual of [729, 587, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    2. linear OA(3¹³⁶, 729, F₃, 35) (dual of [729, 593, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
    3. linear OA(3¹, 7, F₃, 1) (dual of [7, 6, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(154−37, 154, 42899)-Net in Base 3 — Upper bound on s

There is no (117, 154, 42900)-net in base 3, because

1 times m-reduction [i] would yield (117, 153, 42900)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9 991204 365034 073751 490170 889066 104582 186396 778956 203740 206992 529569 394841 > 3¹⁵³ [i]