MinT - Best Known (199−45, 199, s)-Nets in Base 4

Best Known (199−45, 199, s)-Nets in Base 4

(199−45, 199, 1044)-Net over F₄ — Constructive and digital

Digital (154, 199, 1044)-net over F₄, using

1 times m-reduction [i] based on digital (154, 200, 1044)-net over F₄, using
- trace code for nets [i] based on digital (4, 50, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(199−45, 199, 3298)-Net over F₄ — Digital

Digital (154, 199, 3298)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁹⁹, 3298, F₄, 45) (dual of [3298, 3099, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁹⁹, 4096, F₄, 45) (dual of [4096, 3897, 46]-code), using
  - an extension C_e(44) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,44], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 45 [i]

(199−45, 199, 791155)-Net in Base 4 — Upper bound on s

There is no (154, 199, 791156)-net in base 4, because

1 times m-reduction [i] would yield (154, 198, 791156)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 161392 424704 596040 169300 147761 151662 724800 536675 651127 487050 248068 649370 350584 694784 241981 256177 333859 459041 923370 351064 > 4¹⁹⁸ [i]