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We describe algorithms and implementations for linear algebra with dense matrices over F2e for 2 ≤ e ≤ 10. Our main contributions are: (1) a specialisation of precomputation tables to F2e , called Newton-John tables in this work, to avoid scalar multiplications in Gaussian elimination and matrix multiplication, (2) an efficient implementation of Karatsuba-style multiplication for matrices over extension fields of F2 and (3) a description of an open-source library -called M4RIE -providing thedoi:10.1145/2442829.2442838 dblp:conf/issac/Albrecht12 fatcat:c7lwujhbbjgyvjyjnklj3unwm4