Describing Quaternary Codes Using Binary Codes

Binary Codes are studied in information theory, electrical engineering, mathematics and computer science. They are used to design efficient and reliable data transmission methods. Linear codes are easier to deal with compared to nonlinear codes. Certain nonlinear binary codes though contain more codewords than any known linear codes with the same length and minimum distance. These include the Nordstrom-Robinson code, Kerdock, Preparata and Goethals codes. The Kerdock and Preparata codes are formal duals. It was not clear if these codes are duals in some more algebraic sense. Then, It was shown that when the Kerdock and Preparata codes are properly defined, they can be simply constructed as binary images under the Gray map of dual quaternary codes. Decoding codes mentioned is greatly simplified by working in the Z_4-domain, where they are linear. Observing Quaternary codes might lead to better binary codes. Here we define a class of quaternary codes, C(C_1,C_2) giving rise to a fixed pair of binary codes C_1= X (mod 2) and C_2= even words in X mapped coordinatewise to the Z_2 domain for X in C(C_1,C_2). We describe this class using the fixed pair of binary codes {C_1,C_2}.