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.

💡 Research Summary

The paper provides a complete classification of Z₄‑linear extended perfect binary codes, i.e., binary codes of length n = 2ᵏ (k ≥ 4) with distance 4 that can be represented as Gray images of quaternary codes over Z₄. The main result is that for every n = 2ᵏ > 8 there exist exactly ⌊(k + 1)/2⌋ mutually nonequivalent Z₄‑linear extended perfect codes, and each of these codes has a distinct rank (the dimension of the linear span of its binary image).

The authors begin by recalling that many well‑known nonlinear binary codes (Golay, Preparata, Goethals, Delsarte‑Goethals) can be described as Gray images of Z₄‑linear codes, using the mapping 0→00, 1→01, 2→11, 3→10. They note that the classical extended Hamming codes of length > 16 are not Z₄‑linear, which motivates the search for all possible Z₄‑linear extended perfect codes.

The construction hinges on two non‑negative integers r₁ and r₂. For each pair (r₁, r₂) a check matrix A_{r₁,r₂} is formed by listing all distinct columns of the form (1, z) where z runs through the Cartesian product {0,1,2,3}^{r₁} × {0,2}^{r₂}, ordered lexicographically. The quaternary code

C_{r₁,r₂} = { c ∈ Z₄^{n} | A_{r₁,r₂} cᵀ = 0 }

has length n = 4^{r₁}·2^{r₂} = 2^{2r₁+r₂+1}. Theorem 1 proves that C_{r₁,r₂} is a perfect quaternary code, i.e., every non‑zero codeword has Lee weight at least 4. By the Gray map φ, each C_{r₁,r₂} yields an extended perfect binary code of the same length and distance.

To show that the codes for different (r₁, r₂) are pairwise nonequivalent, the authors introduce the Even and Odd operators, which extract the even‑indexed and odd‑indexed columns (or coordinates) of a matrix (or word). Proposition 1 establishes that Even(A_{r₁,r₂}) and Odd(A_{r₁,r₂}) are equivalent to A_{r₁,r₂−1} (when r₂ > 0) or to A_{r₁−1,1} (when r₂ = 0). This recursive relationship allows the authors to analyze the structure of C_{r₁,r₂} by induction, proving that the even and odd subcodes coincide with the codes of lower parameters (Corollary 1).

Rank analysis is performed via the notion of repetitive words (binary words that repeat after n/2 positions) and the observation that φ maps the set {0,2}ⁿ onto the set of repetitive binary words. Proposition 7 shows that the space of repetitive words dual to C_{r₁,r₂} has dimension r₁ + r₂ + 1, leading to the upper bound rank(C_{r₁,r₂}) ≤ n − r₁ − r₂ − 1 (Corollary 2). For the special family C_{0,r₂} (r₂ ≥ 4) the authors prove that the rank attains the bound exactly: rank(C_{0,r₂}) = n − log₂ n (Theorem 8). This demonstrates that these codes are genuinely nonlinear (they cannot be the linear Hamming codes, whose rank would be n − log₂ n − 1).

A concrete example is given for C_{1,1}, whose rank is shown to be 13 by explicitly listing 13 linearly independent binary vectors in its Gray image (Theorem 9). Since each pair (r₁, r₂) yields a different value of r₁ + r₂, the ranks of the corresponding codes are distinct, establishing pairwise nonequivalence.

Section 5 presents an inductive construction (essentially a Mollard‑type recursion) that generates all C_{r₁,r₂} from the base case C_{0,1}. This construction mirrors earlier cyclic Z₄‑linear code descriptions and confirms that no other Z₄‑linear extended perfect codes exist beyond those obtained by the matrix A_{r₁,r₂}.

In summary, the paper proves that for every power‑of‑two length n > 8 there are exactly ⌊(log₂ n + 1)/2⌋ Z₄‑linear extended perfect binary codes, each with a unique rank, and provides explicit constructions, rank calculations, and a proof of completeness. The results close a long‑standing gap in the theory of additive codes, showing precisely how many Z₄‑linear perfect structures exist and how they differ from classical linear Hamming codes. This work lays a solid foundation for further exploration of Z₄‑linear and more general additive code families.