Best Known (62, 62+168, s)-Nets in Base 4
(62, 62+168, 66)-Net over F4 — Constructive and digital
Digital (62, 230, 66)-net over F4, using
- t-expansion [i] based on digital (49, 230, 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
(62, 62+168, 99)-Net over F4 — Digital
Digital (62, 230, 99)-net over F4, using
- t-expansion [i] based on digital (61, 230, 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
(62, 62+168, 318)-Net over F4 — Upper bound on s (digital)
There is no digital (62, 230, 319)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4230, 319, F4, 168) (dual of [319, 89, 169]-code), but
- residual code [i] would yield OA(462, 150, S4, 42), but
- the linear programming bound shows that M ≥ 1 659762 510567 225532 778073 956682 851160 927584 506340 544111 988906 878425 443399 317508 383360 785706 508045 270473 316579 147776 / 70196 277944 430470 237066 876146 512897 904291 630830 313677 137903 195390 483423 677609 > 462 [i]
- residual code [i] would yield OA(462, 150, S4, 42), but
(62, 62+168, 410)-Net in Base 4 — Upper bound on s
There is no (62, 230, 411)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 3 517792 474860 944035 044016 315388 695843 619349 024476 293249 136735 370829 565343 261588 193314 392765 614047 913620 083853 899336 218569 808975 370751 544060 > 4230 [i]