MinT - Best Known (195−43, 195, s)-Nets in Base 4

Best Known (195−43, 195, s)-Nets in Base 4

(195−43, 195, 1048)-Net over F₄ — Constructive and digital

Digital (152, 195, 1048)-net over F₄, using

1 times m-reduction [i] based on digital (152, 196, 1048)-net over F₄, using
- trace code for nets [i] based on digital (5, 49, 262)-net over F₂₅₆, using
  - net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(195−43, 195, 3766)-Net over F₄ — Digital

Digital (152, 195, 3766)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁹⁵, 3766, F₄, 43) (dual of [3766, 3571, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁹⁵, 4111, F₄, 43) (dual of [4111, 3916, 44]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(4¹⁹⁴, 4110, F₄, 43) (dual of [4110, 3916, 44]-code), using
    - constructionÂ X applied to C([0,21]) âŠ‚ C([0,20]) [i] based on
      1. linear OA(4¹⁹³, 4097, F₄, 43) (dual of [4097, 3904, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,21], and minimum distance dÂ â‰¥ |{−21,−20,…,21}|+1Â =Â 44 (BCH-bound) [i]
      2. linear OA(4¹⁸¹, 4097, F₄, 41) (dual of [4097, 3916, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,20], and minimum distance dÂ â‰¥ |{−20,−19,…,20}|+1Â =Â 42 (BCH-bound) [i]
      3. linear OA(4¹, 13, F₄, 1) (dual of [13, 12, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(195−43, 195, 1054998)-Net in Base 4 — Upper bound on s

There is no (152, 195, 1054999)-net in base 4, because

1 times m-reduction [i] would yield (152, 194, 1054999)-net in base 4, but
- 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]