Best Known (65, 242, s)-Nets in Base 4
(65, 242, 66)-Net over F4 — Constructive and digital
Digital (65, 242, 66)-net over F4, using
- t-expansion [i] based on digital (49, 242, 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, 242, 99)-Net over F4 — Digital
Digital (65, 242, 99)-net over F4, using
- t-expansion [i] based on digital (61, 242, 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, 242, 332)-Net over F4 — Upper bound on s (digital)
There is no digital (65, 242, 333)-net over F4, because
- 1 times m-reduction [i] would yield digital (65, 241, 333)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4241, 333, F4, 176) (dual of [333, 92, 177]-code), but
- residual code [i] would yield OA(465, 156, S4, 44), but
- the linear programming bound shows that M ≥ 5305 672794 944537 381320 515598 514645 629008 946704 424695 546266 632663 854579 036277 493526 466626 448910 185539 633024 632652 385646 851129 344000 / 3 530621 465654 281569 250779 664225 038700 823798 252333 134165 234221 216150 895884 348866 248497 833837 > 465 [i]
- residual code [i] would yield OA(465, 156, S4, 44), but
- extracting embedded orthogonal array [i] would yield linear OA(4241, 333, F4, 176) (dual of [333, 92, 177]-code), but
(65, 242, 429)-Net in Base 4 — Upper bound on s
There is no (65, 242, 430)-net in base 4, because
- 1 times m-reduction [i] would yield (65, 241, 430)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 14 326810 286833 536782 667740 753940 213791 543500 645921 428761 527126 866552 609452 911431 832062 235765 881004 357207 431318 836920 904250 902395 053667 736221 191100 > 4241 [i]