# F. Jessie MacWilliams

Florence Jessie Collinson MacWilliams (born 1917 in Stoke-on-Trent ; † May 27, 1990 ) was a British-American mathematician who worked on coding theory .

MacWilliams studied at Cambridge University , where she received her bachelor's degree in 1938 and her master's degree in 1939. She then went on a scholarship to Johns Hopkins University , where she studied with Oscar Zariski , whom she also followed to Harvard in 1940 . In 1941 she married, raised her three children and from 1958 worked as a programmer at Bell Laboratories , where her husband Walter MacWilliams worked as an engineer. In order to become a scientist at Bell Labs, she did her doctorate in 1961 at Harvard (Combinatorial problems of elementary group theory) with Andrew Gleason . Her dissertation was on coding theory (a topic she became interested in after a lecture by RC Bose at Bell Labs) and included the MacWilliams identity of coding theory, later named after her, which combines the weight counting polynomial of a code with that of its dual code connects. In 1983 she retired from Bell Labs.

In 1977 her book with Neil Sloane "The Theory of Error Correcting Codes" was published by North-Holland, an encyclopaedic work with over 1500 references that was very influential in coding theory.

In 1980 she was the first Noether Lecturer . Her daughter Ann also became a mathematician and was even studying at Harvard at the same time her mother was doing her doctorate there.

## Publications

• Combinatorial problems of elementary Abelian groups , Dissertation, 1962
• On the p-rank of the design matrix of a difference set , Madison, Wisconsin: Mathematics Research Center, United States Army, University of Wisconsin, 1967
• The theory of error correcting codes , Amsterdam / New York: North-Holland Publishing Company, 1976, ISBN 0-444-85009-0

## Remarks

1. With the weight counting polynomial of code C (consisting of "words" of length n with letters 0.1), where the number of code words with w is ones, and the dual code says the MacWilliams identity: ${\ displaystyle W (C; x, y) = \ sum _ {w = 0} ^ {n} A_ {w} x ^ {w} y ^ {nw}}$${\ displaystyle A_ {w}}$${\ displaystyle C ^ {\ perp} = \ {x \ in \ mathbb {F} _ {2} ^ {n} \, \ mid \, \ langle x, c \ rangle = 0 {\ mbox {}} \ forall c \ in C \}}$
${\ displaystyle W (C ^ {\ perp}; x, y) = {\ frac {1} {\ mid C \ mid}} W (C; yx, y + x).}$