Best Known (248−179, 248, s)-Nets in Base 4
(248−179, 248, 66)-Net over F4 — Constructive and digital
Digital (69, 248, 66)-net over F4, using
- t-expansion [i] based on digital (49, 248, 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
(248−179, 248, 99)-Net over F4 — Digital
Digital (69, 248, 99)-net over F4, using
- t-expansion [i] based on digital (61, 248, 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
(248−179, 248, 394)-Net over F4 — Upper bound on s (digital)
There is no digital (69, 248, 395)-net over F4, because
- 3 times m-reduction [i] would yield digital (69, 245, 395)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4245, 395, F4, 176) (dual of [395, 150, 177]-code), but
- residual code [i] would yield OA(469, 218, S4, 44), but
- the linear programming bound shows that M ≥ 109109 147274 768315 990931 948474 366203 066087 657712 317992 394776 862369 613323 594430 467474 878725 704856 371200 / 292745 629447 360691 756384 897898 834188 795527 737754 030331 320673 > 469 [i]
- residual code [i] would yield OA(469, 218, S4, 44), but
- extracting embedded orthogonal array [i] would yield linear OA(4245, 395, F4, 176) (dual of [395, 150, 177]-code), but
(248−179, 248, 460)-Net in Base 4 — Upper bound on s
There is no (69, 248, 461)-net in base 4, because
- 1 times m-reduction [i] would yield (69, 247, 461)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 60504 347907 806502 464585 695550 257220 559786 623746 333605 045504 037315 070847 237329 316718 654032 025363 492076 501989 629603 136369 151748 717643 266529 559403 579136 > 4247 [i]