Minimum Weights and Weight Enumerators of $\BBZ_{4}$-Linear Quadratic Residue Codes

A fast method to compute the minimum Lee weight and the symmetrized weight enumerator of extended quadratic residue codes (XQR-codes) over the ring Z
₄
is developed. Our approach is based on the classical Brouwer-Zimmermann algorithm and additionally takes advantage of the large group of automorphisms and the self-duality of the Z
₄
-linear XQR-codes as well as the projection to the binary XQR-codes. As a result, the hitherto unknown minimum Lee distances of all Z
₄
-linear XQR-codes of lengths between 72 and 104 and the minimum Euclidean distances for the lengths 72, 80, and 104 are computed. It turns out that the binary Gray image of the Z
₄
-linear XQR-codes of lengths 80 and 104 has higher minimum distance than any known linear binary code of equal length and cardinality. Furthermore, the Z
₄
-linear XQR-code of length 80 is a new example of an extremal Z
₄
-linear typeII code. Additionally, we give the symmetrized weight enumerator of the Z
₄
-linear XQR-codes of lengths 72 and 80, and we correct the weight enumerators of the Z
₄
-linear XQR-code of length 48 given by Pless and Qian and Bonnecaze et al.