MinT - Best Known (149−35, 149, s)-Nets in Base 3

Best Known (149−35, 149, s)-Nets in Base 3

(149−35, 149, 464)-Net over F₃ — Constructive and digital

Digital (114, 149, 464)-net over F₃, using

3¹ times duplication [i] based on digital (113, 148, 464)-net over F₃, using
- trace code for nets [i] based on digital (2, 37, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(149−35, 149, 854)-Net over F₃ — Digital

Digital (114, 149, 854)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁴⁹, 854, F₃, 35) (dual of [854, 705, 36]-code), using
- 106 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, 9 times 0, 1, 13 times 0, 1, 16 times 0, 1, 19 times 0, 1, 22 times 0) [i] based on linear OA(3¹³⁶, 735, F₃, 35) (dual of [735, 599, 36]-code), using
  - constructionÂ X applied to C_e(34) âŠ‚ C_e(33) [i] based on
    1. 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]
    2. linear OA(3¹³⁰, 729, F₃, 34) (dual of [729, 599, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    3. linear OA(3⁰, 6, F₃, 0) (dual of [6, 6, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(149−35, 149, 51113)-Net in Base 3 — Upper bound on s

There is no (114, 149, 51114)-net in base 3, because

1 times m-reduction [i] would yield (114, 148, 51114)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 41115 139074 044936 540072 527059 507124 506109 486716 755712 772600 569088 369077 > 3¹⁴⁸ [i]