Best Known (228−166, 228, s)-Nets in Base 4
(228−166, 228, 66)-Net over F4 — Constructive and digital
Digital (62, 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
(228−166, 228, 99)-Net over F4 — Digital
Digital (62, 228, 99)-net over F4, using
- t-expansion [i] based on 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
- net from sequence [i] based on digital (61, 98)-sequence over F4, using
(228−166, 228, 333)-Net over F4 — Upper bound on s (digital)
There is no digital (62, 228, 334)-net over F4, because
- 2 times m-reduction [i] would yield digital (62, 226, 334)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4226, 334, F4, 164) (dual of [334, 108, 165]-code), but
- residual code [i] would yield OA(462, 169, S4, 41), but
- the linear programming bound shows that M ≥ 25171 060604 496198 113359 740373 368869 999898 334565 616347 337981 288178 184867 760688 088668 492201 991549 747200 / 1168 180553 676758 470189 656469 231757 605150 850528 809269 341506 088169 > 462 [i]
- residual code [i] would yield OA(462, 169, S4, 41), but
- extracting embedded orthogonal array [i] would yield linear OA(4226, 334, F4, 164) (dual of [334, 108, 165]-code), but
(228−166, 228, 411)-Net in Base 4 — Upper bound on s
There is no (62, 228, 412)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 215754 052146 982562 851115 268435 468410 833344 507003 395543 430603 050594 486274 093762 549227 503865 511941 074559 179996 160847 860515 189992 396047 841010 > 4228 [i]