Best Known (63, 233, s)-Nets in Base 4
(63, 233, 66)-Net over F4 — Constructive and digital
Digital (63, 233, 66)-net over F4, using
- t-expansion [i] based on digital (49, 233, 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
(63, 233, 99)-Net over F4 — Digital
Digital (63, 233, 99)-net over F4, using
- t-expansion [i] based on digital (61, 233, 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
(63, 233, 333)-Net over F4 — Upper bound on s (digital)
There is no digital (63, 233, 334)-net over F4, because
- 2 times m-reduction [i] would yield digital (63, 231, 334)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4231, 334, F4, 168) (dual of [334, 103, 169]-code), but
- residual code [i] would yield OA(463, 165, S4, 42), but
- the linear programming bound shows that M ≥ 34 077190 942729 726882 024839 335902 035522 717143 016283 935337 510998 524998 580004 361097 586089 334532 997120 / 383476 318008 814835 283839 224897 225684 554347 388608 827379 925139 > 463 [i]
- residual code [i] would yield OA(463, 165, S4, 42), but
- extracting embedded orthogonal array [i] would yield linear OA(4231, 334, F4, 168) (dual of [334, 103, 169]-code), but
(63, 233, 416)-Net in Base 4 — Upper bound on s
There is no (63, 233, 417)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 197 195029 786515 525692 972211 707505 145477 180366 232362 842248 148657 429280 957103 442643 154864 668163 992460 195681 134700 869936 103241 161687 166791 532160 > 4233 [i]