$${\mathbb{Z}_2\mathbb{Z}_4}$$ -linear codes: rank and kernel

Abstract

A code C is Z 2 Z 4 -additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C by deleting the coordinates outside X (respectively, Y ) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of Z 2 Z 4 -additive codes under an extended Gray map are called Z 2 Z 4 -linear codes. In this paper, the invariants for Z 2 Z 4 -linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of Z 2 Z 4 -linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a Z 2 Z 4 -linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a Z 2 Z 4 -linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a Z 2 Z 4 -linear code for each possible pair (r, k) is given.

Maximum distance separable codes over $${\mathbb{Z}_4}$$ and $${\mathbb{Z}_2 \times \mathbb{Z}_4}$$

“…For cases (ii) and (iii) in Theorem 4, we apply the result given in [6] which states that the rank of C must be in the range γ + 2δ, . .…”

“…Since 2 (0 1110010) * (0 1001101) / ∈ C 8 , where * denotes the component-wise product, then from [6] the rank is 10 and therefore; C 8 has a binary non-linear image.…”

Kernels and ranks of cyclic and negacyclic quaternary codes

“…It is well known that the kernel of a binary code is the intersection of all maximal linear subspaces and that the code is the union of cosets of the kernel; see [9], [10] for details.…”

“…In [9], [10], various bounds are put on the rank and size of the kernel for arbitrary quaternary codes. In this work, these bounds are significantly refined for the cyclic case.…”

Construction of Hadamard <inline-formula> <tex-math notation="LaTeX">$ {\mathbb {Z}}_{2} {\mathbb {Z}}_{4} {Q}_{8}$ </tex-math></inline-formula>-Codes for Each Allowable Value of the Rank and Dimension of the Kernel

“…On the other hand, from the side of coding theory, it is desirable that the algebraic structures preserve the Hamming distance. This is the case of the Z 2 Z 4 -linear codes that have been intensively studied in recent years [4], [6]. Translation invariant propelinear codes have been characterized as the image of a subgroup by a suitable Gray map of a direct product of Z 2 , Z 4 and Q 8 [10].…”