Best Known (65, 65+166, s)-Nets in Base 4
(65, 65+166, 66)-Net over F4 — Constructive and digital
Digital (65, 231, 66)-net over F4, using
- t-expansion [i] based on digital (49, 231, 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
(65, 65+166, 99)-Net over F4 — Digital
Digital (65, 231, 99)-net over F4, using
- t-expansion [i] based on digital (61, 231, 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
(65, 65+166, 386)-Net over F4 — Upper bound on s (digital)
There is no digital (65, 231, 387)-net over F4, because
- 2 times m-reduction [i] would yield digital (65, 229, 387)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4229, 387, F4, 164) (dual of [387, 158, 165]-code), but
- residual code [i] would yield OA(465, 222, S4, 41), but
- the linear programming bound shows that M ≥ 32 342197 523814 651165 392954 045591 141352 749935 799714 032461 858391 219530 982071 151362 048000 / 23186 656803 327364 360991 616330 320246 466025 092007 > 465 [i]
- residual code [i] would yield OA(465, 222, S4, 41), but
- extracting embedded orthogonal array [i] would yield linear OA(4229, 387, F4, 164) (dual of [387, 158, 165]-code), but
(65, 65+166, 435)-Net in Base 4 — Upper bound on s
There is no (65, 231, 436)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 13 010060 086707 700503 663947 502812 521489 636625 268512 370973 244689 246714 198942 037072 993540 720967 500117 574733 970517 260827 466379 312609 930763 987630 > 4231 [i]