Best Known (246−187, 246, s)-Nets in Base 4
(246−187, 246, 66)-Net over F4 — Constructive and digital
Digital (59, 246, 66)-net over F4, using
- t-expansion [i] based on digital (49, 246, 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
(246−187, 246, 91)-Net over F4 — Digital
Digital (59, 246, 91)-net over F4, using
- t-expansion [i] based on digital (50, 246, 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
(246−187, 246, 245)-Net over F4 — Upper bound on s (digital)
There is no digital (59, 246, 246)-net over F4, because
- 7 times m-reduction [i] would yield digital (59, 239, 246)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4239, 246, F4, 180) (dual of [246, 7, 181]-code), but
- residual code [i] would yield linear OA(459, 65, F4, 45) (dual of [65, 6, 46]-code), but
- 1 times truncation [i] would yield linear OA(458, 64, F4, 44) (dual of [64, 6, 45]-code), but
- residual code [i] would yield linear OA(414, 19, F4, 11) (dual of [19, 5, 12]-code), but
- 1 times truncation [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]
- 1 times truncation [i] would yield linear OA(413, 18, F4, 10) (dual of [18, 5, 11]-code), but
- residual code [i] would yield linear OA(414, 19, F4, 11) (dual of [19, 5, 12]-code), but
- 1 times truncation [i] would yield linear OA(458, 64, F4, 44) (dual of [64, 6, 45]-code), but
- residual code [i] would yield linear OA(459, 65, F4, 45) (dual of [65, 6, 46]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(4239, 246, F4, 180) (dual of [246, 7, 181]-code), but
(246−187, 246, 382)-Net in Base 4 — Upper bound on s
There is no (59, 246, 383)-net in base 4, because
- 1 times m-reduction [i] would yield (59, 245, 383)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 3487 750310 976273 700130 833001 312978 680605 772459 330926 691202 177801 059284 488991 801011 018815 232073 547710 533057 281388 810713 665465 986798 579155 440209 545270 > 4245 [i]