Plotkin construction: rank and kernel

Given two binary codes of length n, using Plotkin construction we obtain a code of length 2n. The construction works for linear and nonlinear codes. For the linear case, it is straightforward to see that the dimension of the final code is the sum of the dimensions of the starting codes. For nonlinear codes, the rank and the dimension of the kernel are standard mesures of linearity. In this report, we prove that both parameters are also the sum of the corresponding ones of the starting codes.

💡 Research Summary

The paper investigates the behavior of two fundamental parameters—rank and kernel dimension—under the Plotkin construction, a classic method for combining two binary codes. Given two binary codes C₁ and C₂ of length n, the Plotkin construction produces a new code C of length 2n defined as
C = {(u, u + v) | u ∈ C₁, v ∈ C₂},
where addition is performed modulo 2. For linear codes the situation is trivial: if C₁ and C₂ have dimensions k₁ and k₂, then C is linear with dimension k₁ + k₂, because the mapping (u, v) ↦ (u, u + v) is a linear isomorphism onto its image and the two subspaces occupy disjoint coordinate blocks.

The novelty of the work lies in extending this additive property to non‑linear codes, where the usual notion of dimension is replaced by two standard measures of linearity: the rank (the dimension of the smallest linear space containing the code) and the kernel dimension (the dimension of the kernel K(C) = { x ∈ 𝔽₂²ⁿ | C + x = C }). The authors prove that both parameters behave additively under the Plotkin construction.

Rank Additivity. Let r₁ and r₂ denote the ranks of C₁ and C₂, respectively. Every codeword of C can be expressed as a linear combination of vectors of the form (u, 0) with u ∈ C₁ and (0, v) with v ∈ C₂. The sets {(u, 0)} and {(0, v)} are linearly independent because their non‑zero entries lie in disjoint halves of the coordinate vector. Consequently the linear span of C is the direct sum of the spans of these two sets, and its dimension is r₁ + r₂. The paper provides a rigorous proof of this independence and shows that no additional linear relations can arise from the mixed term (u, u + v).

Kernel Dimension Additivity. For a vector x = (a, b) to belong to K(C), the condition C + x = C must hold for all (u, u + v) ∈ C. Expanding this requirement yields two simultaneous constraints: a must belong to K(C₁), and b must satisfy b = a + k₂ for some k₂ ∈ K(C₂). Hence the kernel of C can be described as
K(C) = { (a, a + k₂) | a ∈ K(C₁), k₂ ∈ K(C₂) }.
This set is isomorphic to the direct product K(C₁) × K(C₂), and therefore its dimension equals dim K(C₁) + dim K(C₂). The authors formalize this argument, verifying that the described vectors indeed form a subgroup of 𝔽₂²ⁿ and that no other vectors satisfy the kernel condition.

Implications and Applications. The additive behavior of rank and kernel dimension means that designers can precisely control the linearity characteristics of large non‑linear codes by iteratively applying the Plotkin construction. Starting from small seed codes with known rank and kernel, each construction step doubles the length while adding the respective parameters. This property is valuable in contexts where a certain amount of linear structure is required (e.g., for efficient decoding or for cryptographic security) but full linearity is undesirable. Potential applications include the design of non‑linear error‑correcting codes with prescribed algebraic properties, construction of cryptographic primitives such as hash functions that rely on controlled non‑linearity, and multi‑user communication schemes where code orthogonality and kernel size influence interference management.

Conclusion and Future Work. By extending the well‑known dimension‑additivity result from linear to non‑linear codes, the paper fills a gap in the theoretical understanding of the Plotkin construction. The authors suggest several directions for further research: adapting the proofs to q‑ary alphabets, exploring the interaction of rank and kernel with other non‑linear invariants (e.g., non‑linear distance spectra), and investigating whether similar additive phenomena hold for other code combination techniques such as the concatenation or the (u|u+v) construction in more general settings. Overall, the work provides a clear, mathematically rigorous foundation for using the Plotkin construction as a building block in the systematic design of complex binary codes with controllable linearity metrics.