On Z2kZ_{2^k}-Dual Binary Codes

A new generalization of the Gray map is introduced. The new generalization $\Phi: Z_{2^k}^n \to Z_{2}^{2^{k-1}n}$ is connected with the known generalized Gray map $\phi$ in the following way: if we take two dual linear $Z_{2^k}$-codes and construct binary codes from them using the generalizations $\phi$ and $\Phi$ of the Gray map, then the weight enumerators of the binary codes obtained will satisfy the MacWilliams identity. The classes of $Z_{2^k}$-linear Hadamard codes and co-$Z_{2^k}$-linear extended 1-perfect codes are described, where co-$Z_{2^k}$-linearity means that the code can be obtained from a linear $Z_{2^k}$-code with the help of the new generalized Gray map. Keywords: Gray map, Hadamard codes, MacWilliams identity, perfect codes, $Z_{2^k}$-linearityComment: English: 10pp, Russian: 14pp; V.1 title: Z_{2^k}-duality, Z_{2^k}-linear Hadamard codes, and co-Z_{2^k}-linear 1-perfect codes; V.2: revised; V.3: minor revision, references updated, Russian translation adde

On the Existence of Extremal Type II Z2k-Codes

For lengths 8,16 and 24, it is known that there is an extremal Type II Z2k-code for every positive integer k. In this paper, we show that there is an extremal Type II Z2k-code of lengths 32,40,48,56 and 64 for every positive integer k. For length 72, it is also shown that there is an extremal Type II Z4k-code for every positive integer k with k \ge 2.Comment: 29 page

An Upper Bound on the Minimum Weight of Type II \ZZ_{2k}-Codes

In this paper, we give a new upper bound on the minimum Euclidean weight of Type II \ZZ_{2k}-codes and the concept of extremality for the Euclidean weights when $k=3,4,5,6$. Together with the known result, we demonstrate that there is an extremal Type II \ZZ_{2k}-code of length $8m$ $(m \le 8)$ when $k=3,4,5,6$.Comment: 10 pages, 2 table

Extremal Type I Zk\mathbb{Z}_k-codes and kk-frames of odd unimodular lattices

For some extremal (optimal) odd unimodular lattice $L$ in dimensions $12,16,20,28,32,36,40$ and $44$, we determine all integers $k$ such that $L$ contains a $k$-frame. This result yields the existence of an extremal Type I $\mathbb{Z}_{k}$-code of lengths $12,16,20,32,36,40$ and $44$, and a near-extremal Type I $\mathbb{Z}_k$-code of length $28$ for positive integers $k$ with only a few exceptions.Comment: 27 pages. arXiv admin note: substantial text overlap with arXiv:1301.517

On the Existence of Frames of Some Extremal Odd Unimodular Lattices and Self-Dual Zk-Codes

For some extremal (optimal) odd unimodular lattices L in dimensions n=12,16,20,32,36,40 and 44, we determine all positive integers k such that L contains a k-frame. This result yields the existence of an extremal Type I Zk-code of lengths 12,16,20,32,36,40 and 44 and a near-extremal Type I Zk-code of length 28 for positive integers k with only a few exceptions.Comment: 36 pages. arXiv admin note: text overlap with arXiv:1205.694

Decompositions of the Moonshine Module with respect to subVOAs associated to codes over Z2k\Z_{2k}

In this paper, we give decompositions of the moonshine module $V^{\natural}$ with respect to subVOAs associated to extremal Type II codes over $Z_{2k}$ for an integer $k\ge2$. Those subVOAs are isomorphic to the tensor product of 24 copies of the charge conjugation orbifold VOA. Using such decompositions, we obtain some elements of type 4A (k odd) and 2B (k even) of the Monster simple group Aut$(V^{\natural})$.Comment: 16 pages, LaTe

Nonexistence for extremal Type II \ZZ_{2k}-Codes

In this paper, we show that an extremal Type II \ZZ_{2k}-code of length $n$ dose not exist for all sufficiently large $n$ when $k=2,3,4,5,6$.Comment: 8 pages. arXiv admin note: substantial text overlap with arXiv:0906.502

Full characterization of generalized bent functions as (semi)-bent spaces, their dual, and the Gray image

In difference to many recent articles that deal with generalized bent (gbent) functions $f:\mathbb{Z}_2^n \rightarrow \mathbb{Z}_q$ for certain small valued $q\in \{4,8,16 \}$, we give a complete description of these functions for both $n$ even and odd and for any $q=2^k$ in terms of both the necessary and sufficient conditions their component functions need to satisfy. This enables us to completely characterize gbent functions as algebraic objects, namely as affine spaces of bent or semi-bent functions with interesting additional properties, which we in detail describe. We also specify the dual and the Gray image of gbent functions for $q=2^k$. We discuss the subclass of gbent functions which corresponds to relative difference sets, which we call $\mathbb{Z}_q$-bent functions, and point out that they correspond to a class of vectorial bent functions. The property of being $\mathbb{Z}_q$-bent is much stronger than the standard concept of a gbent function. We analyse two examples of this class of functions.Comment: 20 page

On ZpZpkZ_pZ_{p^k}-additive codes and their duality

In this paper, two different Gray-like maps from $Z_p^\alpha\times Z_{p^k}^\beta$, where $p$ is prime, to $Z_p^n$, $n={\alpha+\beta p^{k-1}}$, denoted by $\phi$ and $\Phi$, respectively, are presented. We have determined the connection between the weight enumerators among the image codes under these two mappings. We show that if $C$ is a $Z_p Z_{p^k}$-additive code, and $C^\bot$ is its dual, then the weight enumerators of the image $p$-ary codes $\phi(C)$ and $\Phi(C^\bot)$ are formally dual. This is a partial generalization of [On $Z_{2^k}$-dual binary codes, arXiv:math/0509325], and the result is generalized to odd characteristic $p$ and mixed alphabet. Additionally, a construction of $1$-perfect additive codes in the mixed $Z_p Z_{p^2} ... Z_{p^k}$ alphabet is given

Z4-Linear Perfect Codes

For every $n = 2^k > 8$ there exist exactly $[(k+1)/2]$ mutually nonequivalent $Z_4$-linear extended perfect codes with distance 4. All these codes have different ranks.Comment: 15p. Bibliography update