MinT - Best Known (253−49, 253, s)-Nets in Base 4

Best Known (253−49, 253, s)-Nets in Base 4

(253−49, 253, 1539)-Net over F₄ — Constructive and digital

Digital (204, 253, 1539)-net over F₄, using

t-expansion [i] based on digital (200, 253, 1539)-net over F₄, using
- 5 times m-reduction [i] based on digital (200, 258, 1539)-net over F₄, using
  - trace code for nets [i] based on digital (28, 86, 513)-net over F₆₄, using
    - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(253−49, 253, 10315)-Net over F₄ — Digital

Digital (204, 253, 10315)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁵³, 10315, F₄, 49) (dual of [10315, 10062, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁵³, 16384, F₄, 49) (dual of [16384, 16131, 50]-code), using
  - an extension C_e(48) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,48], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 49 [i]

(253−49, 253, 6852833)-Net in Base 4 — Upper bound on s

There is no (204, 253, 6852834)-net in base 4, because

1 times m-reduction [i] would yield (204, 252, 6852834)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52 374415 009686 871431 298603 284367 213741 532882 145201 701308 826787 139305 596002 601966 517509 788665 368407 946183 052277 369383 344254 479142 063547 122640 550868 326196 > 4²⁵² [i]