Best Known (16, 16+35, s)-Nets in Base 49
(16, 16+35, 66)-Net over F49 — Constructive and digital
Digital (16, 51, 66)-net over F49, using
- net from sequence [i] based on digital (16, 65)-sequence over F49, using
(16, 16+35, 93)-Net over F49 — Digital
Digital (16, 51, 93)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4951, 93, F49, 3, 35) (dual of [(93, 3), 228, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4951, 100, F49, 3, 35) (dual of [(100, 3), 249, 36]-NRT-code), using
- construction X applied to AG(3;F,237P) ⊂ AG(3;F,251P) [i] based on
- linear OOA(4938, 91, F49, 3, 35) (dual of [(91, 3), 235, 36]-NRT-code), using algebraic-geometric NRT-code AG(3;F,237P) [i] based on function field F/F49 with g(F) = 3 and N(F) ≥ 92, using
- linear OOA(4924, 91, F49, 3, 21) (dual of [(91, 3), 249, 22]-NRT-code), using algebraic-geometric NRT-code AG(3;F,251P) [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,237P) ⊂ AG(3;F,251P) [i] based on
- discarding factors / shortening the dual code based on linear OOA(4951, 100, F49, 3, 35) (dual of [(100, 3), 249, 36]-NRT-code), using
(16, 16+35, 13983)-Net in Base 49 — Upper bound on s
There is no (16, 51, 13984)-net in base 49, because
- 1 times m-reduction [i] would yield (16, 50, 13984)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 3 237243 560040 344812 157541 070163 699371 145493 453683 860296 200078 818101 687611 850083 311105 > 4950 [i]