MinT - Best Known (248−184, 248, s)-Nets in Base 4

Best Known (248−184, 248, s)-Nets in Base 4

(248−184, 248, 66)-Net over F₄ — Constructive and digital

Digital (64, 248, 66)-net over F₄, using

t-expansion [i] based on digital (49, 248, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(248−184, 248, 99)-Net over F₄ — Digital

Digital (64, 248, 99)-net over F₄, using

t-expansion [i] based on digital (61, 248, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(248−184, 248, 281)-Net over F₄ — Upper bound on s (digital)

There is no digital (64, 248, 282)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁸, 282, F₄, 184) (dual of [282, 34, 185]-code), but
- residual code [i] would yield OA(4⁶⁴, 97, S₄, 46), but
  - the linear programming bound shows that MÂ â‰¥ 17 551672 443269 830978 915105 133499 110631 230918 120257 846151 906745 909248 / 44694 499640 915429 319268 220967 > 4⁶⁴ [i]

(248−184, 248, 418)-Net in Base 4 — Upper bound on s

There is no (64, 248, 419)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 239626 262459 295973 758759 925570 800878 834928 454772 007684 795335 186931 727850 167987 418115 036405 215478 569998 208592 851123 601124 636263 419131 567244 718697 933840 > 4²⁴⁸ [i]