Best Known (61, 61+167, s)-Nets in Base 4
(61, 61+167, 66)-Net over F4 — Constructive and digital
Digital (61, 228, 66)-net over F4, using
- t-expansion [i] based on digital (49, 228, 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
(61, 61+167, 99)-Net over F4 — Digital
Digital (61, 228, 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
(61, 61+167, 319)-Net over F4 — Upper bound on s (digital)
There is no digital (61, 228, 320)-net over F4, because
- 3 times m-reduction [i] would yield digital (61, 225, 320)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4225, 320, F4, 164) (dual of [320, 95, 165]-code), but
- residual code [i] would yield OA(461, 155, S4, 41), but
- the linear programming bound shows that M ≥ 35 162266 348965 864492 465363 741528 241521 515995 482555 223491 846236 883313 207330 736210 116983 129154 519040 / 6 178619 757792 266774 478203 941876 542733 832278 327110 913669 619231 > 461 [i]
- residual code [i] would yield OA(461, 155, S4, 41), but
- extracting embedded orthogonal array [i] would yield linear OA(4225, 320, F4, 164) (dual of [320, 95, 165]-code), but
(61, 61+167, 403)-Net in Base 4 — Upper bound on s
There is no (61, 228, 404)-net in base 4, because
- 1 times m-reduction [i] would yield (61, 227, 404)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 52500 719087 102455 387342 841227 595162 671401 055054 096946 990812 774704 075468 767908 613754 552297 280066 181931 839861 115722 003496 546500 485589 351494 > 4227 [i]