Best Known (235−176, 235, s)-Nets in Base 4
(235−176, 235, 66)-Net over F4 — Constructive and digital
Digital (59, 235, 66)-net over F4, using
- t-expansion [i] based on digital (49, 235, 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
(235−176, 235, 91)-Net over F4 — Digital
Digital (59, 235, 91)-net over F4, using
- t-expansion [i] based on digital (50, 235, 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
(235−176, 235, 246)-Net over F4 — Upper bound on s (digital)
There is no digital (59, 235, 247)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4235, 247, F4, 176) (dual of [247, 12, 177]-code), but
- construction Y1 [i] would yield
- linear OA(4234, 241, F4, 176) (dual of [241, 7, 177]-code), but
- residual code [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
- residual code [i] would yield linear OA(458, 64, F4, 44) (dual of [64, 6, 45]-code), but
- OA(412, 247, 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(4234, 241, F4, 176) (dual of [241, 7, 177]-code), but
- construction Y1 [i] would yield
(235−176, 235, 384)-Net in Base 4 — Upper bound on s
There is no (59, 235, 385)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 3275 648035 901720 743032 449837 189302 469285 083737 771291 996685 757745 150625 329467 578107 760030 573763 531504 371856 823525 538055 439919 538856 420332 909200 > 4235 [i]