Study on the algebraic decoding of the (89, 45, 17) quadratic residue code

A study on the weight-5 error patterns of the algebraic decoding of the (89, 45, 17) quadratic residue (QR) code with reducible generator polynomial, proposed by Truong et al. (2008), is presented. For the QR codes with irreducible generator polynomial, S₁ = 0 means that the received word has no errors. For the (89, 45, 17) QR codes, some weight-5 error patterns will cause a zero value in syndrome S₁. However, these weight-5 error patterns do not cause a decoding failure while using the inverse-free Berlekamp-Massey (IFBM) algorithm to determine the error-locator polynomial. An example shows the facts.