Best Known (231−170, 231, s)-Nets in Base 4
(231−170, 231, 66)-Net over F4 — Constructive and digital
Digital (61, 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
(231−170, 231, 99)-Net over F4 — Digital
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
(231−170, 231, 303)-Net over F4 — Upper bound on s (digital)
There is no digital (61, 231, 304)-net over F4, because
- 2 times m-reduction [i] would yield digital (61, 229, 304)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4229, 304, F4, 168) (dual of [304, 75, 169]-code), but
- residual code [i] would yield OA(461, 135, S4, 42), but
- the linear programming bound shows that M ≥ 14553 096315 947253 640884 776913 168926 302451 665455 919283 135527 723519 103281 609358 744533 636652 915559 579757 398043 602406 873493 852833 510337 625815 402107 895808 / 2665 663747 457635 051830 020094 359849 599949 322414 129592 914794 116842 536378 425306 982781 909625 123448 656307 846295 612863 > 461 [i]
- residual code [i] would yield OA(461, 135, S4, 42), but
- extracting embedded orthogonal array [i] would yield linear OA(4229, 304, F4, 168) (dual of [304, 75, 169]-code), but
(231−170, 231, 401)-Net in Base 4 — Upper bound on s
There is no (61, 231, 402)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 13 287158 371701 115913 179698 182096 654606 820035 928555 899253 516503 059500 447704 821506 774063 403387 486130 096879 818468 069153 572573 089902 550084 508448 > 4231 [i]