Best Known (207−148, 207, s)-Nets in Base 4
(207−148, 207, 66)-Net over F4 — Constructive and digital
Digital (59, 207, 66)-net over F4, using
- t-expansion [i] based on digital (49, 207, 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
(207−148, 207, 91)-Net over F4 — Digital
Digital (59, 207, 91)-net over F4, using
- t-expansion [i] based on digital (50, 207, 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
(207−148, 207, 357)-Net over F4 — Upper bound on s (digital)
There is no digital (59, 207, 358)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4207, 358, F4, 148) (dual of [358, 151, 149]-code), but
- residual code [i] would yield OA(459, 209, S4, 37), but
- the linear programming bound shows that M ≥ 3446 699662 509398 967731 365644 377802 984275 634503 925011 721737 994240 000000 000000 / 10213 023598 987442 955247 667333 699389 861739 > 459 [i]
- residual code [i] would yield OA(459, 209, S4, 37), but
(207−148, 207, 398)-Net in Base 4 — Upper bound on s
There is no (59, 207, 399)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 43503 865281 383604 865464 923485 642113 488118 508063 709307 399425 153628 369012 475870 857438 936238 946987 619712 891247 470129 250311 024114 > 4207 [i]