Sign codes from number fields
Three new families of binary linear codes are created from a number field. They are created by looking at the “signs” of the embedding of the units into the real numbers, the 2-adic numbers and then a combination of the two embeddings. The transmission rates of these codes are described in terms of the class group of the number field. Then lower bounds are found for the transmission rates of the codes. In the case of the real sign code, heuristic results of Cohen and Lenstra are used to determine the probability of certain rates occurring. A property of the minimum distance is discussed and a relationship between the minimum distances of the three codes is proven.