MinT - Best Known (80, 80+29, s)-Nets in Base 4

Best Known (80, 80+29, s)-Nets in Base 4

(80, 80+29, 531)-Net over F₄ — Constructive and digital

Digital (80, 109, 531)-net over F₄, using

4¹ times duplication [i] based on digital (79, 108, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 36, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(80, 80+29, 912)-Net over F₄ — Digital

Digital (80, 109, 912)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁰⁹, 912, F₄, 29) (dual of [912, 803, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁰⁹, 1037, F₄, 29) (dual of [1037, 928, 30]-code), using
  - constructionÂ X applied to C_e(28) âŠ‚ C_e(25) [i] based on
    1. linear OA(4¹⁰⁶, 1024, F₄, 29) (dual of [1024, 918, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    2. linear OA(4⁹⁶, 1024, F₄, 26) (dual of [1024, 928, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
    3. linear OA(4³, 13, F₄, 2) (dual of [13, 10, 3]-code), using
      - discarding factors / shortening the dual code based on linear OA(4³, 21, F₄, 2) (dual of [21, 18, 3]-code), using
        Hamming code H(3,4) [i]

(80, 80+29, 88867)-Net in Base 4 — Upper bound on s

There is no (80, 109, 88868)-net in base 4, because

1 times m-reduction [i] would yield (80, 108, 88868)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 105314 655770 871323 649265 346633 312749 323308 913358 796600 745052 928680 > 4¹⁰⁸ [i]