Best Known (218−163, 218, s)-Nets in Base 4
(218−163, 218, 66)-Net over F4 — Constructive and digital
Digital (55, 218, 66)-net over F4, using
- t-expansion [i] based on digital (49, 218, 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
(218−163, 218, 91)-Net over F4 — Digital
Digital (55, 218, 91)-net over F4, using
- t-expansion [i] based on digital (50, 218, 91)-net over F4, using
- net from sequence [i] based on digital (50, 90)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 50 and N(F) ≥ 91, using
- net from sequence [i] based on digital (50, 90)-sequence over F4, using
(218−163, 218, 237)-Net over F4 — Upper bound on s (digital)
There is no digital (55, 218, 238)-net over F4, because
- 3 times m-reduction [i] would yield digital (55, 215, 238)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4215, 238, F4, 160) (dual of [238, 23, 161]-code), but
- construction Y1 [i] would yield
- linear OA(4214, 226, F4, 160) (dual of [226, 12, 161]-code), but
- construction Y1 [i] would yield
- linear OA(4213, 220, F4, 160) (dual of [220, 7, 161]-code), but
- residual code [i] would yield linear OA(453, 59, F4, 40) (dual of [59, 6, 41]-code), but
- residual code [i] would yield linear OA(413, 18, F4, 10) (dual of [18, 5, 11]-code), but
- “Liz†bound on codes from Brouwer’s database [i]
- residual code [i] would yield linear OA(413, 18, F4, 10) (dual of [18, 5, 11]-code), but
- residual code [i] would yield linear OA(453, 59, F4, 40) (dual of [59, 6, 41]-code), but
- OA(412, 226, S4, 6), but
- discarding factors would yield OA(412, 156, S4, 6), but
- the Rao or (dual) Hamming bound shows that M ≥ 16 866019 > 412 [i]
- discarding factors would yield OA(412, 156, S4, 6), but
- linear OA(4213, 220, F4, 160) (dual of [220, 7, 161]-code), but
- construction Y1 [i] would yield
- OA(423, 238, S4, 12), but
- discarding factors would yield OA(423, 205, S4, 12), but
- the Rao or (dual) Hamming bound shows that M ≥ 70 503355 038244 > 423 [i]
- discarding factors would yield OA(423, 205, S4, 12), but
- linear OA(4214, 226, F4, 160) (dual of [226, 12, 161]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(4215, 238, F4, 160) (dual of [238, 23, 161]-code), but
(218−163, 218, 360)-Net in Base 4 — Upper bound on s
There is no (55, 218, 361)-net in base 4, because
- 1 times m-reduction [i] would yield (55, 217, 361)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 52770 919438 455661 239041 764861 341895 189521 386508 961359 114036 225388 101124 411544 430320 951516 124017 971600 376184 907543 502032 862543 258920 > 4217 [i]