MinT - Best Known (204−43, 204, s)-Nets in Base 4

Best Known (204−43, 204, s)-Nets in Base 4

(204−43, 204, 1060)-Net over F₄ — Constructive and digital

Digital (161, 204, 1060)-net over F₄, using

trace code for nets [i] based on digital (8, 51, 265)-net over F₂₅₆, using
- net from sequence [i] based on digital (8, 264)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(204−43, 204, 4648)-Net over F₄ — Digital

Digital (161, 204, 4648)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁰⁴, 4648, F₄, 43) (dual of [4648, 4444, 44]-code), using
- 535 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 24 times 0, 1, 44 times 0, 1, 71 times 0, 1, 101 times 0, 1, 125 times 0, 1, 141 times 0) [i] based on linear OA(4¹⁹³, 4102, F₄, 43) (dual of [4102, 3909, 44]-code), using
  - constructionÂ X applied to C_e(42) âŠ‚ C_e(41) [i] based on
    1. linear OA(4¹⁹³, 4096, F₄, 43) (dual of [4096, 3903, 44]-code), using an extension C_e(42) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,42], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 43 [i]
    2. 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]
    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]

(204−43, 204, 1911088)-Net in Base 4 — Upper bound on s

There is no (161, 204, 1911089)-net in base 4, because

1 times m-reduction [i] would yield (161, 203, 1911089)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 165 265702 044932 483165 841354 344350 465293 745088 780995 962486 224378 625372 121629 744613 095481 505360 168645 731846 492770 425395 807168 > 4²⁰³ [i]