MinT - Best Known (194−42, 194, s)-Nets in Base 4

Best Known (194−42, 194, s)-Nets in Base 4

(194−42, 194, 1052)-Net over F₄ — Constructive and digital

Digital (152, 194, 1052)-net over F₄, using

4² times duplication [i] based on digital (150, 192, 1052)-net over F₄, using
- trace code for nets [i] based on digital (6, 48, 263)-net over F₂₅₆, using
  - net from sequence [i] based on digital (6, 262)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(194−42, 194, 4151)-Net over F₄ — Digital

Digital (152, 194, 4151)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁹⁴, 4151, F₄, 42) (dual of [4151, 3957, 43]-code), using
- 42 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) [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]

(194−42, 194, 1054998)-Net in Base 4 — Upper bound on s

There is no (152, 194, 1054999)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 630 438030 227181 361005 056124 196559 441338 369099 093548 512233 164188 129717 260786 789483 680243 861516 752304 487484 193672 679438 > 4¹⁹⁴ [i]