Best Known (240−171, 240, s)-Nets in Base 4
(240−171, 240, 66)-Net over F4 — Constructive and digital
Digital (69, 240, 66)-net over F4, using
- t-expansion [i] based on digital (49, 240, 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
(240−171, 240, 99)-Net over F4 — Digital
Digital (69, 240, 99)-net over F4, using
- t-expansion [i] based on digital (61, 240, 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
(240−171, 240, 433)-Net over F4 — Upper bound on s (digital)
There is no digital (69, 240, 434)-net over F4, because
- 3 times m-reduction [i] would yield digital (69, 237, 434)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4237, 434, F4, 168) (dual of [434, 197, 169]-code), but
- residual code [i] would yield OA(469, 265, S4, 42), but
- the linear programming bound shows that M ≥ 52348 985175 743312 195164 699043 560255 242722 252419 549522 172173 323504 377479 561128 758476 800000 / 141682 475264 470819 104457 406563 443902 424070 593403 > 469 [i]
- residual code [i] would yield OA(469, 265, S4, 42), but
- extracting embedded orthogonal array [i] would yield linear OA(4237, 434, F4, 168) (dual of [434, 197, 169]-code), but
(240−171, 240, 466)-Net in Base 4 — Upper bound on s
There is no (69, 240, 467)-net in base 4, because
- 1 times m-reduction [i] would yield (69, 239, 467)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 887857 393674 429091 833849 993016 010791 734744 281558 068070 767900 318879 334216 123255 210057 081180 361932 711682 762417 565644 364352 015990 169564 221017 990720 > 4239 [i]