MinT - Best Known (198−42, 198, s)-Nets in Base 4

Best Known (198−42, 198, s)-Nets in Base 4

(198−42, 198, 1056)-Net over F₄ — Constructive and digital

Digital (156, 198, 1056)-net over F₄, using

4² times duplication [i] based on digital (154, 196, 1056)-net over F₄, using
- trace code for nets [i] based on digital (7, 49, 264)-net over F₂₅₆, using
  - net from sequence [i] based on digital (7, 263)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(198−42, 198, 4417)-Net over F₄ — Digital

Digital (156, 198, 4417)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁹⁸, 4417, F₄, 42) (dual of [4417, 4219, 43]-code), using
- 304 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 0, 1, 5 times 0, 1, 11 times 0, 1, 19 times 0, 1, 33 times 0, 1, 52 times 0, 1, 74 times 0, 1, 99 times 0) [i] based on linear OA(4¹⁸⁷, 4102, F₄, 42) (dual of [4102, 3915, 43]-code), using
  - constructionÂ X applied to C_e(41) âŠ‚ C_e(40) [i] based on
    1. linear OA(4¹⁸⁷, 4096, F₄, 42) (dual of [4096, 3909, 43]-code), using an extension C_e(41) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,41], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 42 [i]
    2. linear OA(4¹⁸¹, 4096, F₄, 41) (dual of [4096, 3915, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [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]

(198−42, 198, 1373825)-Net in Base 4 — Upper bound on s

There is no (156, 198, 1373826)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 161391 867711 235640 725332 107189 769989 947863 559811 681407 875980 760276 109640 567306 795747 072501 089915 003764 433283 951631 304864 > 4¹⁹⁸ [i]