Abstract
It is shown thatthe residue code of a self-dual $\mathbb{Z}_4$-code of length $24k$(resp. $24k+8$)and minimum Lee weight $8k+4 \text{ or }8k+2$(resp. $8k+8 \text{ or }8k+6$)is a binary extremal doubly even self-dual codefor every positive integer $k$.A number of new self-dual $\mathbb{Z}_4$-codes of length $24$and minimum Lee weight $10$ are constructed using theabove characterization.These codes are Type I $\mathbb{Z}_4$-codeshaving the largest minimum Lee weight and the largestEuclidean weight among all Type I $\mathbb{Z}_4$-codes of that length.In addition, new extremal Type II $\mathbb{Z}_4$-codes of length$56$ are found.
Highlights
Self-dual codes are an important class of codes1 for both theoretical and practical reasons
Among self-dual Zk-codes, self-dual Z4-codes have been widely studied because such codes have nice applications to unimodular lattices and binary codes, where Zk denotes the ring of integers modulo k and k is a positive integer with k ≥ 2
The Nordstorm–Robinson and Kerdock codes, which are some of the best known non-linear binary codes, can be constructed as the Gray images of some Z4-codes [8]
Summary
Self-dual codes are an important class of (linear) codes for both theoretical and practical reasons. The Nordstorm–Robinson and Kerdock codes, which are some of the best known non-linear binary codes, can be constructed as the Gray images of some Z4-codes [8]. We pay attention to the minimum Lee weight from the viewpoint of a connection with the minimum distance of binary (non-linear) codes obtained as the Gray images. It is shown that the residue code of a selfdual Z4-code of length 24k and minimum Lee weight 8k + 4 or 8k + 2 is a binary extremal doubly even self-dual code of length 24k for every positive integer k. It is shown that the residue code of a self-dual Z4-code of length 24k + 8 and minimum Lee weight 8k + 8 or 8k + 6 is a binary extremal doubly even self-dual code of length 24k + 8. All computer calculations in this note were done by Magma [4]
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