Properties of Codes from Difference Sets in 2-Groups

• Authors: Deirdre Longacher Smeltzer
• Venue: Designs, Codes and Cryptography
• Year: 1999

A( v, k, \lambda)-differenceset D in a group G can be used to createa symmetric 2-( v, k, \lambda)design, \cal D, from which arises a code C,generated by vectors corresponding to the characteristic functionof blocks of \cal D. This paper examines propertiesof the code C, and of a subcode, C^o=JC,where J is the radical of the group algebra of Gover {\Bbb Z}_2. When G is a 2-group,it is shown that C^o is equivalent to the first-orderReed-Muller code, {\cal R}(1, 2s+2), precisely whenthe 2-divisor of C^o is maximal. In addition, ifD is a non-trivial difference set in an elementaryabelian 2-group, and if D is generated by a quadraticbent function, then C^o is equal to a power of theradical. Finally, an example is given of a difference set whosecharacteristic function is not quadratic, although the 2-divisorof C^o is maximal.