Best Known (17, 17+37, s)-Nets in Base 49
(17, 17+37, 96)-Net over F49 — Constructive and digital
Digital (17, 54, 96)-net over F49, using
- net from sequence [i] based on digital (17, 95)-sequence over F49, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) ≤ 17 and N(F) ≥ 96, using
- F4 from the tower of function fields by GarcÃa, Stichtenoth, and Rück over F49 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) ≤ 17 and N(F) ≥ 96, using
(17, 17+37, 97)-Net over F49 — Digital
Digital (17, 54, 97)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4954, 97, F49, 3, 37) (dual of [(97, 3), 237, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4954, 100, F49, 3, 37) (dual of [(100, 3), 246, 38]-NRT-code), using
- strength reduction [i] based on linear OOA(4954, 100, F49, 3, 38) (dual of [(100, 3), 246, 39]-NRT-code), using
- construction X applied to AG(3;F,234P) ⊂ AG(3;F,248P) [i] based on
- linear OOA(4941, 91, F49, 3, 38) (dual of [(91, 3), 232, 39]-NRT-code), using algebraic-geometric NRT-code AG(3;F,234P) [i] based on function field F/F49 with g(F) = 3 and N(F) ≥ 92, using
- linear OOA(4927, 91, F49, 3, 24) (dual of [(91, 3), 246, 25]-NRT-code), using algebraic-geometric NRT-code AG(3;F,248P) [i] based on function field F/F49 with g(F) = 3 and N(F) ≥ 92 (see above)
- linear OOA(4913, 9, F49, 3, 13) (dual of [(9, 3), 14, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4913, 49, F49, 3, 13) (dual of [(49, 3), 134, 14]-NRT-code), using
- Reed–Solomon NRT-code RS(3;134,49) [i]
- discarding factors / shortening the dual code based on linear OOA(4913, 49, F49, 3, 13) (dual of [(49, 3), 134, 14]-NRT-code), using
- construction X applied to AG(3;F,234P) ⊂ AG(3;F,248P) [i] based on
- strength reduction [i] based on linear OOA(4954, 100, F49, 3, 38) (dual of [(100, 3), 246, 39]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4954, 100, F49, 3, 37) (dual of [(100, 3), 246, 38]-NRT-code), using
(17, 17+37, 14904)-Net in Base 49 — Upper bound on s
There is no (17, 54, 14905)-net in base 49, because
- 1 times m-reduction [i] would yield (17, 53, 14905)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 380740 996876 796503 209211 455236 393645 797793 724738 202885 284562 370547 370866 195354 960233 747809 > 4953 [i]