Best Known (51, 51+173, s)-Nets in Base 2
(51, 51+173, 36)-Net over F2 — Constructive and digital
Digital (51, 224, 36)-net over F2, using
- net from sequence [i] based on digital (51, 35)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 41, N(F) = 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F2 with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
(51, 51+173, 40)-Net over F2 — Digital
Digital (51, 224, 40)-net over F2, using
- t-expansion [i] based on digital (50, 224, 40)-net over F2, using
- net from sequence [i] based on digital (50, 39)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 50 and N(F) ≥ 40, using
- net from sequence [i] based on digital (50, 39)-sequence over F2, using
(51, 51+173, 66)-Net in Base 2 — Upper bound on s
There is no (51, 224, 67)-net in base 2, because
- 31 times m-reduction [i] would yield (51, 193, 67)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2193, 67, S2, 3, 142), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 2 008672 555323 737844 427452 615426 453253 152753 742228 491044 126720 / 143 > 2193 [i]
- extracting embedded OOA [i] would yield OOA(2193, 67, S2, 3, 142), but