The weight enumerator polynomial is a mathematical tool that encapsulates the weight distribution of a linear code, detailing how many codewords exist for each possible weight. It helps in understanding the error-correcting capabilities of the code by providing insights into the number of codewords with specific weights, which are crucial for decoding and performance analysis. This polynomial plays a significant role in characterizing codes and is instrumental when discussing concepts like dual codes and the MacWilliams identity, which relates the weight enumerator of a code to that of its dual.
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