Best Known (85, s)-Sequences in Base 2
(85, 51)-Sequence over F2 — Constructive and digital
Digital (85, 51)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 3 places with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
(85, 56)-Sequence over F2 — Digital
Digital (85, 56)-sequence over F2, using
- t-expansion [i] based on digital (83, 56)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 83 and N(F) ≥ 57, using
(85, 93)-Sequence in Base 2 — Upper bound on s
There is no (85, 94)-sequence in base 2, because
- net from sequence [i] would yield (85, m, 95)-net in base 2 for arbitrarily large m, but
- m-reduction [i] would yield (85, 656, 95)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2656, 95, S2, 7, 571), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 48 139661 438954 597921 667719 398539 061055 180622 375617 701401 280511 697340 550474 668248 873530 291822 401735 580998 789486 434628 375479 640487 371829 773449 211312 372139 719675 646518 399598 766126 749630 133154 821435 293696 / 143 > 2656 [i]
- extracting embedded OOA [i] would yield OOA(2656, 95, S2, 7, 571), but
- m-reduction [i] would yield (85, 656, 95)-net in base 2, but