Best Known (220−159, 220, s)-Nets in Base 4
(220−159, 220, 66)-Net over F4 — Constructive and digital
Digital (61, 220, 66)-net over F4, using
- t-expansion [i] based on digital (49, 220, 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
(220−159, 220, 99)-Net over F4 — Digital
Digital (61, 220, 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
(220−159, 220, 356)-Net over F4 — Upper bound on s (digital)
There is no digital (61, 220, 357)-net over F4, because
- 3 times m-reduction [i] would yield digital (61, 217, 357)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4217, 357, F4, 156) (dual of [357, 140, 157]-code), but
- residual code [i] would yield OA(461, 200, S4, 39), but
- the linear programming bound shows that M ≥ 2 016494 570162 278735 518158 542717 648360 131511 752234 027270 947252 517739 366352 683008 / 364228 998441 829554 049476 381326 982618 927461 > 461 [i]
- residual code [i] would yield OA(461, 200, S4, 39), but
- extracting embedded orthogonal array [i] would yield linear OA(4217, 357, F4, 156) (dual of [357, 140, 157]-code), but
(220−159, 220, 408)-Net in Base 4 — Upper bound on s
There is no (61, 220, 409)-net in base 4, because
- 1 times m-reduction [i] would yield (61, 219, 409)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 826916 113012 326310 597210 057322 090800 609643 732963 318087 298361 023086 204494 188544 471358 687520 343521 926055 138598 136812 686690 478896 691552 > 4219 [i]