Best Known (219−164, 219, s)-Nets in Base 4
(219−164, 219, 66)-Net over F4 — Constructive and digital
Digital (55, 219, 66)-net over F4, using
- t-expansion [i] based on digital (49, 219, 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
(219−164, 219, 91)-Net over F4 — Digital
Digital (55, 219, 91)-net over F4, using
- t-expansion [i] based on digital (50, 219, 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
(219−164, 219, 230)-Net over F4 — Upper bound on s (digital)
There is no digital (55, 219, 231)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4219, 231, F4, 164) (dual of [231, 12, 165]-code), but
- construction Y1 [i] would yield
- linear OA(4218, 225, F4, 164) (dual of [225, 7, 165]-code), but
- residual code [i] would yield linear OA(454, 60, F4, 41) (dual of [60, 6, 42]-code), but
- 1 times truncation [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
- 1 times truncation [i] would yield linear OA(453, 59, F4, 40) (dual of [59, 6, 41]-code), but
- residual code [i] would yield linear OA(454, 60, F4, 41) (dual of [60, 6, 42]-code), but
- OA(412, 231, 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(4218, 225, F4, 164) (dual of [225, 7, 165]-code), but
- construction Y1 [i] would yield
(219−164, 219, 359)-Net in Base 4 — Upper bound on s
There is no (55, 219, 360)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 779162 918007 731891 803517 420353 433826 105051 046463 776838 004042 601778 345599 562068 975652 482732 598345 423746 629047 857431 717323 365896 872122 > 4219 [i]