Z2Z4-linear codes: generator matrices and duality

A code ${\cal C}$ 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 ${\cal C}$ by deleting the coordinates outside $X$ (respectively, $Y$) is a binary linear code (respectively, a quaternary linear code). In this paper $\Z_2\Z_4$-additive codes are studied. Their corresponding binary images, via the Gray map, are $\Z_2\Z_4$-linear codes, which seem to be a very distinguished class of binary group codes. As for binary and quaternary linear codes, for these codes the fundamental parameters are found and standard forms for generator and parity check matrices are given. For this, the appropriate inner product is deduced and the concept of duality for $\Z_2\Z_4$-additive codes is defined. Moreover, the parameters of the dual codes are computed. Finally, some conditions for self-duality of $\Z_2\Z_4$-additive codes are given.

💡 Research Summary

The paper introduces and thoroughly investigates ℤ₂ℤ₄‑additive codes, a class of group codes whose coordinate set can be split into two disjoint subsets X and Y such that the puncturing of the code on X yields a binary linear code while puncturing on Y yields a quaternary linear code. By applying the Gray map φ, which expands each ℤ₄‑coordinate into a pair of binary coordinates (0→00, 1→01, 2→11, 3→10), every ℤ₂ℤ₄‑additive code 𝒞 is transformed into a binary code φ(𝒞). This binary image, called a ℤ₂ℤ₄‑linear code, inherits the Lee distance of 𝒞 as a Hamming distance, making it a natural extension of both binary and quaternary linear codes.

The authors first formalize the fundamental parameters of an ℤ₂ℤ₄‑additive code. If α denotes the number of binary (ℤ₂) coordinates and β the number of quaternary (ℤ₄) coordinates, then the code can be described by a 4‑tuple (α, β; γ, δ). Here γ is the rank of the binary part and δ the rank of the quaternary part; consequently the size of the code is |𝒞| = 2^γ · 4^δ. The length of the binary image is n = α + 2β, and the minimum distance can be expressed in terms of the Lee distance of 𝒞.

A central contribution of the work is the derivation of canonical forms for generator and parity‑check matrices. The generator matrix G is shown to be equivalent, under coordinate permutations and elementary row operations, to a block matrix of the form

  G =