Minimum weights and weight enumerators of ℤ₄-linear quadratic residue codes

Title data

Kiermaier, Michael ; Wassermann, Alfred:
Minimum weights and weight enumerators of ℤ₄-linear quadratic residue codes.
In: IEEE Transactions on Information Theory. Vol. 58 (2012) Issue 7 . - pp. 4870-4883.
ISSN 0018-9448
DOI: https://doi.org/10.1109/TIT.2012.2191389

Project information

Abstract in another language

A fast method to compute the minimum Lee weight and the symmetrized weight enumerator of extended quadratic residue codes (XQR-codes) over the ring ℤ₄ 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 ℤ₄-linear XQR-codes as well as the projection to the binary XQR-codes.

As a result, the hitherto unknown minimum Lee distances of all ℤ₄-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 ℤ₄-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 ℤ₄-linear XQR-code of length 80 is a new example of an extremal ℤ₄-linear type II code. Additionally, we give the symmetrized weight enumerator of the ℤ₄-linear XQR-codes of lengths 72 and 80, and we correct the weight enumerators of the ℤ₄-linear XQR-code of length 48 given by Pless and Qian and Bennecaze et al.