Z2Z4-linear codes: rank and kernel

A code C is Z2Z4-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). In this paper, the rank and dimension of the kernel for Z2Z4-linear codes, which are the corresponding binary codes of Z2Z4-additive codes, are studied. The possible values of these two parameters for Z2Z4-linear codes, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a Z2Z4-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a Z2Z4-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 Z2Z4-additive code for each possible pair (r,k) is given.

💡 Research Summary

The paper investigates two fundamental structural parameters of Z₂Z₄‑linear codes: the rank (the dimension of the binary linear span) and the kernel dimension (the size of the translation‑invariant subcode). A Z₂Z₄‑additive code 𝒞 is defined as a subgroup of Z₂^α × Z₄^β; applying the Gray map Φ yields a binary code C = Φ(𝒞), called a Z₂Z₄‑linear code. The authors introduce a compact type notation (α, β; γ, δ; κ) where γ and δ count the number of order‑2 and order‑4 generators, respectively, and κ is the dimension of the binary part of the order‑2 subcode.

The first major contribution is a complete description of all possible rank values for a given type. By analyzing the generator matrix in a canonical form (equation 3) and using Lemma 3, the authors show that the binary span h(C) is generated by three families of vectors: Φ(u_i) (order‑2 generators), Φ(v_j) together with Φ(2v_j) (order‑4 generators), and the products Φ(2v_j ∗ v_k) for 1 ≤ j < k ≤ δ. The first two families always contribute γ + 2δ independent vectors; the third family can add at most ⌊δ/2⌋ further independent vectors, but is also limited by the number of binary coordinates β and by κ. Consequently the rank satisfies

 γ + 2δ ≤ rank(C) ≤ min(β + δ + κ, γ + 2δ + ⌊δ/2⌋).

Proposition 1 proves that every integer r within this interval is attainable. The construction proceeds by selecting a suitable subset of the product vectors Φ(2v_j ∗ v_k) and inserting them as additional rows in the generator matrix, thereby raising the rank by exactly the desired amount. The authors illustrate the tightness of the bounds with known families: extended 1‑perfect Z₄‑linear codes (which achieve the upper bound) and Hadamard Z₄‑linear codes (which occupy intermediate values).

The second major contribution concerns the kernel K(C). By definition K(C) = {x ∈ Z₂ⁿ | C + x = C}. The paper shows that its dimension k obeys

 κ ≤ k ≤ γ + δ,

where κ is the binary dimension of the order‑2 subcode and γ + δ is the total number of independent generators of 𝒞. Moreover, when k is fixed, the admissible rank values shrink to

 γ + 2δ ≤ rank(C) ≤ γ + 2δ + min(β − γ + κ, ⌊δ/2⌋).

The authors construct, for each feasible pair (r, k), a Z₂Z₄‑additive code whose Gray image has exactly those parameters. The method starts from a canonical generator matrix, then adjusts κ by adding or removing order‑2 rows in the binary part, and finally tunes the rank by the controlled inclusion of product rows Φ(2v_j ∗ v_k).

The paper also discusses implementation aspects: the authors have added a dedicated package to MAGMA that supports Z₂Z₄‑additive codes, including functions to compute rank, kernel, and to generate the families described.

In summary, the work provides a full characterization of the rank‑kernel landscape for Z₂Z₄‑linear codes, establishes tight lower and upper bounds, and supplies explicit constructions achieving every admissible combination. This advances the theoretical understanding of these hybrid codes and offers practical tools for designing codes with prescribed linear‑span and translation‑invariance properties, which may be valuable in applications such as modulation schemes, cryptographic primitives, and the study of other non‑linear code families.