Best Known (246−181, 246, s)-Nets in Base 4
(246−181, 246, 66)-Net over F4 — Constructive and digital
Digital (65, 246, 66)-net over F4, using
- t-expansion [i] based on digital (49, 246, 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
(246−181, 246, 99)-Net over F4 — Digital
Digital (65, 246, 99)-net over F4, using
- t-expansion [i] based on digital (61, 246, 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
(246−181, 246, 316)-Net over F4 — Upper bound on s (digital)
There is no digital (65, 246, 317)-net over F4, because
- 1 times m-reduction [i] would yield digital (65, 245, 317)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4245, 317, F4, 180) (dual of [317, 72, 181]-code), but
- residual code [i] would yield OA(465, 136, S4, 45), but
- the linear programming bound shows that M ≥ 415878 730529 031135 021679 860495 672856 565078 457128 582331 480821 965858 300674 152452 442836 617388 483062 874023 037003 308108 967020 842896 601894 883099 015808 041663 566504 021681 595442 145183 008509 397628 248876 052647 239899 999262 067613 171313 541120 / 282 957338 342515 520783 463442 333260 804660 722953 291161 529077 196208 799173 693527 287967 908946 060987 265800 746662 194089 764699 465545 861267 440205 293981 112010 916143 596828 540138 272039 423935 111737 383517 > 465 [i]
- residual code [i] would yield OA(465, 136, S4, 45), but
- extracting embedded orthogonal array [i] would yield linear OA(4245, 317, F4, 180) (dual of [317, 72, 181]-code), but
(246−181, 246, 427)-Net in Base 4 — Upper bound on s
There is no (65, 246, 428)-net in base 4, because
- 1 times m-reduction [i] would yield (65, 245, 428)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 3660 156352 604094 041428 853306 673220 888809 264115 080419 022967 542960 629645 507108 245996 213073 667524 922862 841949 637188 759989 871532 715641 275228 660683 987740 > 4245 [i]