MinT - Best Known (140, 178, s)-Nets in Base 4

Best Known (140, 178, s)-Nets in Base 4

(140, 178, 1052)-Net over F₄ — Constructive and digital

Digital (140, 178, 1052)-net over F₄, using

4² times duplication [i] based on digital (138, 176, 1052)-net over F₄, using
- trace code for nets [i] based on digital (6, 44, 263)-net over F₂₅₆, using
  - net from sequence [i] based on digital (6, 262)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(140, 178, 4192)-Net over F₄ — Digital

Digital (140, 178, 4192)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁷⁸, 4192, F₄, 38) (dual of [4192, 4014, 39]-code), using
- 81 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0, 1, 19 times 0, 1, 31 times 0) [i] based on linear OA(4¹⁶⁹, 4102, F₄, 38) (dual of [4102, 3933, 39]-code), using
  - constructionÂ X applied to C_e(37) âŠ‚ C_e(36) [i] based on
    1. linear OA(4¹⁶⁹, 4096, F₄, 38) (dual of [4096, 3927, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
    2. linear OA(4¹⁶³, 4096, F₄, 37) (dual of [4096, 3933, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [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]

(140, 178, 1154622)-Net in Base 4 — Upper bound on s

There is no (140, 178, 1154623)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 146785 339204 635636 380003 613759 619609 252924 457353 742528 936341 699619 828893 406619 819920 665707 061941 136913 454376 > 4¹⁷⁸ [i]