Best Known (247−182, 247, s)-Nets in Base 4
(247−182, 247, 66)-Net over F4 — Constructive and digital
Digital (65, 247, 66)-net over F4, using
- t-expansion [i] based on digital (49, 247, 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
(247−182, 247, 99)-Net over F4 — Digital
Digital (65, 247, 99)-net over F4, using
- t-expansion [i] based on digital (61, 247, 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
(247−182, 247, 316)-Net over F4 — Upper bound on s (digital)
There is no digital (65, 247, 317)-net over F4, because
- 2 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
(247−182, 247, 426)-Net in Base 4 — Upper bound on s
There is no (65, 247, 427)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 57433 193897 426821 014307 726196 285867 875759 005052 390005 188241 518470 610635 287526 205990 758977 104913 464120 428492 872500 750166 644897 187274 047001 107667 656880 > 4247 [i]