Best Known (63, 63+175, s)-Nets in Base 4
(63, 63+175, 66)-Net over F4 — Constructive and digital
Digital (63, 238, 66)-net over F4, using
- t-expansion [i] based on digital (49, 238, 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+175, 99)-Net over F4 — Digital
Digital (63, 238, 99)-net over F4, using
- t-expansion [i] based on digital (61, 238, 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+175, 317)-Net over F4 — Upper bound on s (digital)
There is no digital (63, 238, 318)-net over F4, because
- 3 times m-reduction [i] would yield digital (63, 235, 318)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4235, 318, F4, 172) (dual of [318, 83, 173]-code), but
- residual code [i] would yield OA(463, 145, S4, 43), but
- the linear programming bound shows that M ≥ 410852 595144 601422 661736 194475 352548 234222 916103 520325 789662 026727 829058 017732 594422 665880 071589 138518 417655 328617 787562 131456 / 4712 214848 200787 196384 454868 078661 057064 361851 730871 257084 069591 637575 047639 691128 659913 > 463 [i]
- residual code [i] would yield OA(463, 145, S4, 43), but
- extracting embedded orthogonal array [i] would yield linear OA(4235, 318, F4, 172) (dual of [318, 83, 173]-code), but
(63, 63+175, 414)-Net in Base 4 — Upper bound on s
There is no (63, 238, 415)-net in base 4, because
- 1 times m-reduction [i] would yield (63, 237, 415)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 50681 877550 478256 005465 263498 352232 994262 398264 343302 162589 892282 989813 441890 599446 618723 416353 197282 623008 843605 065115 679281 017517 124319 256584 > 4237 [i]