MinT - Best Known (145−33, 145, s)-Nets in Base 4

Best Known (145−33, 145, s)-Nets in Base 4

(145−33, 145, 1040)-Net over F₄ — Constructive and digital

Digital (112, 145, 1040)-net over F₄, using

4¹ times duplication [i] based on digital (111, 144, 1040)-net over F₄, using
- trace code for nets [i] based on digital (3, 36, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(145−33, 145, 2567)-Net over F₄ — Digital

Digital (112, 145, 2567)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁴⁵, 2567, F₄, 33) (dual of [2567, 2422, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁴⁵, 4096, F₄, 33) (dual of [4096, 3951, 34]-code), using
  - an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]

(145−33, 145, 594221)-Net in Base 4 — Upper bound on s

There is no (112, 145, 594222)-net in base 4, because

1 times m-reduction [i] would yield (112, 144, 594222)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 497 336320 619442 932462 927554 189217 635631 933019 043297 649859 356971 490228 432248 846929 586762 > 4¹⁴⁴ [i]