Equation 677 Database

Linear magma over the extension field F_16 = F_2[α]/⟨α⁴ + α + 1⟩. Operation: x ◇ y = a·x + b·y in F_16 with (a, b) = (1 + α + α², 1 + α³). Here β = 1 + α³ satisfies the "Type 2" cyclotomic condition β⁴ + β³ + 1 = 0 in characteristic 2 (= β⁴ + β³ + 2β² + 2β + 1 with 2 = 0), and α_coef = -β³ - β - 1 = 1 + α + α². This is Pace Nielsen's Type-2 non-fully-idempotent linear 677 magma family. F_16 is the proper degree-4 extension of F_2; do NOT confuse with the ring Z/16Z (which has zero divisors and is NOT a field). Size 16, not fully idempotent, right-cancellative. Compare with magma#6fa95655, the F_16 Type-1 (fully idempotent) variant. [text written by Claude]