## A Survey of Coding Theory

### San Antonio, Texas 1980

**FLORENCE JESSIE COLLINSON MACWILLIAMS** was born in 1917 in Stoke-on-Trent, England. She received her BA in 1938 and her MA the following year, both from Cambridge University. In 1939, she received a traveling scholarship from Cambridge and went to Johns Hopkins University, where she studied with Oscar Zariski. In 1940, she followed Zariski to Harvard University to study there for a year. She married in 1941 and left her mathematical work for some years to raise her three children, one daughter and two sons.

In 1958, MacWilliams went to work as a computer programmer at Bell Telephone Laboratories in Murray Hill, New Jersey, where her husband; Walter MacWilliams, had been hired as an engineer after the war. She became interested in coding theory when R. C. Bose came to Bell Labs and gave a talk on the subject. MacWilliams wanted to become a member of the Bell Labs technical staff, a position requiring a PhD, so in 1961, she returned to Harvard for a year and obtained a PhD, studying coding theory with Andrew Gleason. (Her daughter Ann, who also has a PhD in mathematics, was studying at Harvard at the same time.) According to an obituary which was written by Vera Pless of the University of Illinois at Chicago and which appeared in *SIAM News* in November 1990, MacWilliams' PhD thesis, "Combinatorial Problems of Elementary Group Theory", contains "one of the most powerful theorems in coding theory."

"The MacWilliams equations relate the weight distribution of a linear code to the weight distribution of its dual code," writes Pless. "When the code equals its dual, it is called self-dual, and these equations tell much about the weight distribution of self-dual codes. The MacWilliams equations are widely used by coding theorists, both to obtain new theoretical information about error-correcting codes and to determine the weight distributions of specific codes." The MacWilliams equations also led to important results in the area of combinatorial designs.

MacWilliams also worked on cyclic codes, generalizing them to abelian group codes. With H. B. Mann, she solved a difficult problem involving certain design matrices. She is perhaps best known for her book, *The Theory of Error-Correcting Codes*, North Holland, 1977), written in collaboration with Neil J.A. Sloane of Bell Labs. In the obituary, Pless notes that this encyclopedic book, with nearly 1,500 references, covers many diverse areas of coding theory. "The many research problems scattered throughout the book have stimulated work in many areas of coding theory," writes Pless.

MacWilliams retired from Bell Labs in January 1983 and devoted herself to her grandchildren, home, and garden. She died in May 1990.