Best Known (222−159, 222, s)-Nets in Base 4
(222−159, 222, 66)-Net over F4 — Constructive and digital
Digital (63, 222, 66)-net over F4, using
- t-expansion [i] based on digital (49, 222, 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
(222−159, 222, 99)-Net over F4 — Digital
Digital (63, 222, 99)-net over F4, using
- t-expansion [i] based on digital (61, 222, 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
(222−159, 222, 389)-Net over F4 — Upper bound on s (digital)
There is no digital (63, 222, 390)-net over F4, because
- 3 times m-reduction [i] would yield digital (63, 219, 390)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4219, 390, F4, 156) (dual of [390, 171, 157]-code), but
- residual code [i] would yield OA(463, 233, S4, 39), but
- the linear programming bound shows that M ≥ 2 394144 335707 471520 884224 011051 775831 492539 148504 246999 784192 681791 545828 966400 000000 / 26753 789254 137656 696530 296222 884590 520869 895903 > 463 [i]
- residual code [i] would yield OA(463, 233, S4, 39), but
- extracting embedded orthogonal array [i] would yield linear OA(4219, 390, F4, 156) (dual of [390, 171, 157]-code), but
(222−159, 222, 424)-Net in Base 4 — Upper bound on s
There is no (63, 222, 425)-net in base 4, because
- 1 times m-reduction [i] would yield (63, 221, 425)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 11 759224 423103 657942 799561 477884 579335 856662 867681 726896 592524 699674 501665 779227 439486 116951 237672 799723 622021 591434 063041 212886 671168 > 4221 [i]