Cosets of Extended BCH Codes of Various Weights

To control transmission errors in a data communication system, one can use a code in three ways: purely for error detection, purely for error correction, or for a combination of error correction and detection. Error correcting codes have been widely studied in these decades, e.g. see McWilliam and Sloane [1977]. The family of triple-error-correcting binary BCH codes of length n=2m-1 has been thoroughly studied since 1960's. The weight distribution of these codes was determined by Kasami [1971] for odd m. For even m, method of Kasami didn't work. Berlekamp [1967, 1978] gave the weight distribution of extended codes for even values of m.