The sequence of nitrogen bases in a DNA double helix primarily carries information for how to make the protein. The specific arrangement of the nitrogen bases (adenine, thymine, cytosine, and guanine) encodes the genetic instructions that direct the synthesis of proteins through the processes of transcription and translation.
While the other options you mentioned (patterns of sugars, partnering codons and anti-codons, and hydrogen bonds) are related to DNA and protein synthesis in some way, they do not directly represent what is carried in the sequence of nitrogen bases.
To summarize, the correct answer is that the sequence of nitrogen bases in DNA carries the information necessary for protein synthesis.