MinT - Best Known (137, 172, s)-Nets in Base 4

Best Known (137, 172, s)-Nets in Base 4

(137, 172, 1061)-Net over F₄ — Constructive and digital

Digital (137, 172, 1061)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 32, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (105, 140, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 35, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(137, 172, 5034)-Net over F₄ — Digital

Digital (137, 172, 5034)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁷², 5034, F₄, 35) (dual of [5034, 4862, 36]-code), using
- 917 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 17 times 0, 1, 28 times 0, 1, 44 times 0, 1, 66 times 0, 1, 94 times 0, 1, 123 times 0, 1, 151 times 0, 1, 174 times 0, 1, 191 times 0) [i] based on linear OA(4¹⁵⁷, 4102, F₄, 35) (dual of [4102, 3945, 36]-code), using
  - constructionÂ X applied to C_e(34) âŠ‚ C_e(33) [i] based on
    1. linear OA(4¹⁵⁷, 4096, F₄, 35) (dual of [4096, 3939, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
    2. linear OA(4¹⁵¹, 4096, F₄, 34) (dual of [4096, 3945, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    3. linear OA(4⁰, 6, F₄, 0) (dual of [6, 6, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(137, 172, 2721679)-Net in Base 4 — Upper bound on s

There is no (137, 172, 2721680)-net in base 4, because

1 times m-reduction [i] would yield (137, 171, 2721680)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 959008 341628 151484 670932 072111 333320 496266 848824 499297 376423 246730 618510 586952 737113 779003 167488 908024 > 4¹⁷¹ [i]