Best Known (63, 63+166, s)-Nets in Base 4
(63, 63+166, 66)-Net over F4 — Constructive and digital
Digital (63, 229, 66)-net over F4, using
- t-expansion [i] based on digital (49, 229, 66)-net over F4, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction 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]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
(63, 63+166, 99)-Net over F4 — Digital
Digital (63, 229, 99)-net over F4, using
- t-expansion [i] based on digital (61, 229, 99)-net over F4, using
- net from sequence [i] based on digital (61, 98)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 61 and N(F) ≥ 99, using
- net from sequence [i] based on digital (61, 98)-sequence over F4, using
(63, 63+166, 349)-Net over F4 — Upper bound on s (digital)
There is no digital (63, 229, 350)-net over F4, because
- 2 times m-reduction [i] would yield digital (63, 227, 350)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4227, 350, F4, 164) (dual of [350, 123, 165]-code), but
- residual code [i] would yield OA(463, 185, S4, 41), but
- the linear programming bound shows that M ≥ 61 899957 150953 106035 135074 637879 046641 791483 715273 894502 688876 097802 046438 481549 501189 324800 / 720746 857989 042001 933213 860469 455977 050620 069897 655507 > 463 [i]
- residual code [i] would yield OA(463, 185, S4, 41), but
- extracting embedded orthogonal array [i] would yield linear OA(4227, 350, F4, 164) (dual of [350, 123, 165]-code), but
(63, 63+166, 419)-Net in Base 4 — Upper bound on s
There is no (63, 229, 420)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 865655 047635 322824 696006 959109 688483 632183 159736 809583 234790 400797 776698 711714 360090 615052 394027 483319 670340 414898 070787 763612 561593 985604 > 4229 [i]