Best Known (69, 69+190, s)-Nets in Base 4
(69, 69+190, 66)-Net over F4 — Constructive and digital
Digital (69, 259, 66)-net over F4, using
- t-expansion [i] based on digital (49, 259, 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
(69, 69+190, 99)-Net over F4 — Digital
Digital (69, 259, 99)-net over F4, using
- t-expansion [i] based on digital (61, 259, 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
(69, 69+190, 349)-Net over F4 — Upper bound on s (digital)
There is no digital (69, 259, 350)-net over F4, because
- 2 times m-reduction [i] would yield digital (69, 257, 350)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4257, 350, F4, 188) (dual of [350, 93, 189]-code), but
- residual code [i] would yield OA(469, 161, S4, 47), but
- 1 times truncation [i] would yield OA(468, 160, S4, 46), but
- the linear programming bound shows that M ≥ 10178 629433 929836 486543 068800 140163 269042 043780 022134 664338 507769 510819 622533 364638 543087 235674 689592 586232 641534 482804 572425 994198 736437 248000 / 113203 572801 045878 145860 058818 437218 882917 411800 440768 742206 357256 206420 022337 868833 760142 804084 942919 > 468 [i]
- 1 times truncation [i] would yield OA(468, 160, S4, 46), but
- residual code [i] would yield OA(469, 161, S4, 47), but
- extracting embedded orthogonal array [i] would yield linear OA(4257, 350, F4, 188) (dual of [350, 93, 189]-code), but
(69, 69+190, 453)-Net in Base 4 — Upper bound on s
There is no (69, 259, 454)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1 007897 068205 513173 938209 265293 853336 796931 071892 856675 018824 745112 389142 251159 196602 784949 113891 769319 008040 294053 347569 700081 308375 075975 919440 950158 896832 > 4259 [i]