Crooked Preparata-like Codes

• Crooked Preparata-like codes are nonlinear binary codes constructed using APN crooked functions over 𝔽₂ with distinct combinatorial constraints.
• They partition the binary Hamming code into additive translates, which induce line-parallelisms and spreads in projective spaces PG(n,2).
• These codes generalize classical Preparata codes, offering deep insights into coding theory, finite geometry, and cryptographic applications.

A crooked Preparata-like code is a nonlinear binary code constructed using a crooked function on a vector space over 𝔽₂. These codes, which exist only for even dimension $n$, always appear as subcodes of the binary linear Hamming code of the same length. Their defining property is that they partition the ambient Hamming code into additive translates, and this structure is leveraged to obtain line-parallelisms in projective spaces PG($n$,2). Crooked Preparata-like codes generalize classical Preparata codes and are constructed using functions that are not only almost perfect nonlinear (APN) but also satisfy additional combinatorial constraints. This article synthesizes their rigorous definition, structural properties, geometric and algebraic implications, and classification, relying primarily on the recent findings of (Heering et al., 27 Aug 2025).

1. Formal Definition and Fundamental Properties

Let $f:\mathbb{F}_{2}^{n} \to \mathbb{F}_{2}^{n}$ be a crooked function; that is,

• $f(0) = 0$,
• for any three distinct elements $x,y,z \in \mathbb{F}_{2}^{n}$,

$f(x) + f(y) + f(z) + f(x+y+z) \ne 0,$

• for any $a\ne 0$ and any $x,y,z \in \mathbb{F}_{2}^{n}$,

$$f(x) + f(y) + f(z) + f(x+a) + f(y+a) + f(z+a) \ne 0.$$

These properties imply $f$ is APN and, in fact, bijective. A Preparata-like code $P_f$ associated with $f$ is defined via sets $X \subset \mathbb{F}_2^n\setminus\{0\}$ and $Y\subset \mathbb{F}_2^n$ such that:

• $|Y|$ is even,
• $\sum_{x\in X} x = \sum_{y\in Y} y,$
• $$\sum_{x\in X} f(x) = \sum_{y\in Y} f(y) + f\Big(\sum_{y\in Y} y\Big).$$

The codewords of $P_f$ are characteristic vectors $(\mathbb{1}_X, \mathbb{1}_Y)$ satisfying the above. It has length $2^n-1$, size $2^{2^n-2n}$, and minimum distance $5$.

2. Partitioning the Hamming Code and Induced Parallelisms

Preparata-like codes are always contained in the binary linear Hamming code $H_n$. Theorem 1 of (Heering et al., 27 Aug 2025) proves that $H_n$ can be partitioned into additive translates of any Preparata-like code $P_f$:

$H_n = \bigsqcup_{a\in A} (P_f + a),$

where the index set $A$ consists of representatives of the cosets.

This partition property yields geometric consequences: considering the minimum weight-3 codewords in the Hamming code as the lines of PG($n$,2), these translates of $P_f$ partition the set of lines into spreads. Collectively, these spreads cover all lines with no overlap, establishing a line-parallelism in PG($n$,2).

A coloring function $c_f$ is defined for pairs of points $(x,x_1), (y,y_1)$ in PG($n$,2) (identified with $\mathbb{F}_2^n \times \mathbb{F}_2$) by

$$c_f((x,x_1),(y,y_1)) = f(x+y) + f(x) + f(y) + f(x_1 y + y_1 x),$$

and determines a partition of the set of lines into color classes.

3. Algebraic and Geometric Equivalence Criteria

The induced line-parallelism (the partition into spreads) depends on the equivalence class of the crooked function used. Theorem 3 from (Heering et al., 27 Aug 2025) states:

• If $f,f'$ are linearly equivalent via linear permutations $L_1, L_2$ such that $f'(x) = L_1(f(L_2(x)))$, then the parallelisms $\Pi_f$ and $\Pi_{f'}$ are equivalent under a collineation (an automorphism of PG($n$,2)).
• For quadratic crooked functions and $n > 3$, affine equivalence is both necessary and sufficient for the equivalence of the induced parallelisms:

$$\Pi_f \cong \Pi_{f'} \iff \exists\ \alpha\in \mathbb{F}_2^n,\ L_1,\ L_2\text{ linear, s.t.}\ f'(x) = L_1(f(L_2(x) + \alpha)) + L_1(f(\alpha)).$$

Thus, classifying such parallelisms reduces to classifying the underlying crooked functions up to affine equivalence, a deep problem in finite field function theory.

4. Generalization of Baker–van Lint–Wilson Construction

Prior work by Baker, van Lint, and Wilson established that generalized Preparata codes partition the Hamming code and induce parallelisms. (Heering et al., 27 Aug 2025) generalizes this result: any Preparata-like code (including the class defined by crooked functions) suffices to partition $H_n$ and to yield such geometric structures. The explicit description of the resulting parallelism via a coloring function allows for systematic construction and classification.

This result creates infinite families of inequivalent parallelisms arising from the plethora of known crooked and APN functions over finite fields.

5. Applications in Coding Theory, Finite Geometry, and Cryptography

The partitioning property of crooked Preparata-like codes leads to several significant implications:

• In coding theory: The partitioning introduces new families of high-minimum-distance nonlinear codes within linear Hamming codes, allowing code designers to exploit both nonlinear protection and regular algebraic structure. The existence of such codes not included in perfect codes also suggests methods for constructing perfect codes free of certain nonlinear subcodes (Krotov et al., 2015).
• In finite geometry: The associated line-parallelisms are highly structured incidence geometries, relevant for symmetric design constructions and for resolvable Steiner systems in affine and projective spaces.
• In cryptography: Crooked (and more generally APN) functions are fundamental for S-box construction in block ciphers. The geometric and coding-theoretic structures arising from crooked Preparata-like codes may inspire algebraic cryptographic primitives or enrich existing ones by constraining possible attack surfaces linked to code equivalence.

Equivalence results for minimal distance graphs and code isometries (0902.1351, 0902.2316) indicate that structural classification via isometry or weak isometry can be extended to crooked Preparata-like codes, provided they share key design-theoretic properties.

6. Structural Constraints and Open Directions

The automorphism group of Preparata-like codes and their extensions—specifically in $Z_2Z_4$-linear context (Krotov, 2016)—exhibit rigidity for codes of length exceeding 16: all symmetries must preserve the underlying algebraic structure, implying that geometric and combinatorial symmetries coincide. This aligns with the equivalence criteria for parallelisms and suggests a rich interplay between affine function theory, nonlinear coding, and finite geometry.

A plausible implication is that the ongoing classification and construction of new crooked functions (especially quadratic ones) will generate further inequivalent families of line-parallelisms and Preparata-like codes, potentially yielding new coding-theoretic and geometric objects with independent interest.

7. Concluding Remarks

Crooked Preparata-like codes represent a generalization of Preparata code methodology via the use of crooked functions on vector spaces over $\mathbb{F}_2$. Their principal feature—the partitioning of the binary Hamming code into additive translates—induces canonical line-parallelisms in projective space PG($n$,2), with the equivalence of these parallelisms directly linked to the classification of crooked functions under affine transformations. These codes connect the algebraic theory of APN and crooked functions with finite geometric constructions, nonlinear coding paradigms, and cryptographic applications, and they generalize the scope of classical results on partitioning and parallelisms in both coding and geometric theory (Heering et al., 27 Aug 2025).