Preparata-like Codes in Modern Coding Theory

• Preparata-like codes are nonlinear binary codes with optimal combinatorial and algebraic structures, enabling minimum distance and weight designs not achievable by comparable linear codes.
• They are deeply connected to finite geometry, with minimum-weight codewords forming t-designs and inducing structured spreads in projective spaces.
• Their hidden Z4-linear structure supports efficient decoding algorithms and inspires advanced quantum error correction strategies with higher performance.

Preparata-like codes are nonlinear binary codes distinguished by their combinatorial optimality, symmetry properties, connections to finite geometry, and utility in quantum error correction and modern coding theory. Foundationally, these codes exhibit minimum distance and weight design properties unattainable by most linear codes of comparable length and cardinality. Their algebraic and geometric structure enables not only deep theoretical characterization but also advanced practical applications in classical and quantum information processing.

1. Algebraic and Combinatorial Structure of Preparata-like Codes

Preparata-like codes typically have length $n = 2^m$ (or $2^m - 1$, or $4^k - 1$ for some integer $k$), cardinality $2^{2^m-2m}$ (or variants depending on the construction), and minimum distance $d = 5$ (punctured codes) or $d = 6$ (extended codes). The codes are strongly distance-invariant and possess a weight distribution that enables their minimum-weight codewords to form $t$–designs, e.g., $3$–$(n,6,(n-4)/3)$ for weight-6 codewords in extended Preparata codes, and $2$–$(n,5,(n-3)/3)$ for weight-5 codewords in punctured versions (0902.2316).

For given $m$ (even, $m \geq 4$) the codewords of minimum weight are highly regular, forming a structure that allows embedding into perfect codes and connections to Steiner systems. Uniquely, Preparata-like codes are nonlinear yet possess a hidden linear structure when viewed via the Gray map as $Z_4$-linear codes.

2. Geometric and Design-Theoretic Connections

Preparata-like codes are intimately connected to finite geometry, especially projective spaces $PG(n,2)$. When embedded in the binary linear Hamming code $H_n$, they partition the code into additive translates, which in turn induce line-parallelisms in projective geometry (Heering et al., 27 Aug 2025). The minimum-weight codewords (weight 3 in $H_n$) correspond to lines in $PG(n-1,2)$, and their organization through code partitions yields spreads and parallelisms. These geometric structures are further elucidated when Preparata-like codes are constructed via crooked functions, which are special almost perfect nonlinear (APN) mappings over $GF(2^n)$. The induced coloring function $c_f$ partitions lines of $PG(n,2)$ into spreads, with equivalence of induced parallelisms tightly controlled by affine equivalence of the underlying crooked functions.

The following table summarizes some parameters:

Code Type	Length $n$	Minimum Distance $d$	Key Design
Preparata-like (unpunctured)	$2^m$	6	$3$–design
Preparata-like (punctured)	$2^m-1$	5	$2$–design
Hamming code	$2^m-1$	3	Steiner STS

The geometric embedding and partitioning mechanisms provide tools for producing new classes of line-parallelisms, as well as applications in design theory and cryptography.

3. Equivalence, Isometries, and Symmetry Groups

Preparata-like codes exhibit a high degree of metric rigidity. For sufficiently large $n$ (e.g., $n \geq 2^{12}$ for full Preparata codes), two codes are weakly isometric (via minimal distance graphs) if and only if they are equivalent (coordinate permutations and translations in $E^n$) (0902.2316). The minimal distance graph, with vertices as codewords and edges linking pairs at distance 6 (extended codes) or 5 (punctured), acts as a complete invariant; isomorphism of such graphs implies full code equivalence (0902.1351).

Automorphism groups of $Z_2Z_4$-linear Preparata-like codes are fully determined by the underlying $Z_2Z_4$ structure when $n > 16$, and every symmetry must commute with the defining involution $T$ (Krotov, 2016). Such rigidity simplifies code classification and ensures the preservation of key combinatorial features.

4. Quantum Codes Derived from Preparata-like Codes

A significant development is the extension of the stabilizer formalism to construct non-additive quantum codes from nonlinear classical codes such as Preparata and Goethals codes (0801.2144, 0801.2150). The construction utilizes the partitioning of nonlinear Preparata-like codes into cosets of a linear Reed–Muller code $R(m)$ to form union stabilizer codes:

$C = \bigcup_{t \in T_0} t C_0,$

where $C_0$ is an additive (stabilizer) code and $T_0$ indexes coset representatives. For Preparata codes, $K_P = 2^{2^m - m + 1}$ cosets are used. The resulting quantum code has parameters $((2^m, K_P 2^k, 6))$, attaining a dimension that surpasses any additive code of the same length and minimum distance. For Quantum Goethals-Preparata codes, the dimension is $2^{2^m-5m+1}$ and the minimum distance is 8 (0801.2150).

These constructions provide higher encoding rates and minimum distances, offering quantum codes with superior performance for fault-tolerant quantum computing and quantum memory architectures. Union stabilizer codes generalize the CSS construction, enabling the realization of non-additive codes with parameters not accessible via conventional linear methods.

5. Decoding and Algorithmic Advances

Efficient and practical decoding of Preparata-like codes is enabled via their quaternary (Gray-mapped $Z_4$-linear) structure (Minja et al., 2023). Maximum a posteriori (MAP) decoders leveraging log-likelihood polynomials and fast transforms (Walsh–Hadamard, Fourier over $Z_4$) achieve $O(N^2 \log_2 N)$ complexity, which is notably lower than exhaustive or trellis-based methods. Sub-optimal bitwise APP decoders capable of working with likelihoods rather than hard decisions exploit dyadic expansion and have complexity $O(N\log_2 N)$. These decoders outperform classical lifting and Chase approaches, achieving error rates within 1.5 dB of the MAP bound in simulation studies.

The dyadic expansion of codewords facilitates the staged decoding: first recovering a "base layer" using SISO algorithms, then sequentially lifting this decision to decode higher-order bits via residue computations and bitwise operations. The generality of these procedures extends to other $Z_4$-codes, ensuring wide applicability in communications.

6. Preparata-like Codes in the Wider Spectrum of Perfect Codes

Although Preparata codes are always embedded in some perfect codes of length $n = 4^k - 1$, the majority of perfect codes (especially those constructed by component switching in Hamming codes via Vasil’ev’s construction) do not contain Preparata-like codes (Krotov et al., 2015). The inclusion of a Preparata code imposes strong structural constraints, such as the linearity of associated Steiner triple systems among codewords of minimal weight. This underscores the combinatorial diversity among perfect codes and motivates further research into the classification and functional properties of codes in this regime.

7. Characteristic Properties, Transitivity, and Regularity

Preparata-like codes, such as the Nordstrom–Robinson code (the smallest member of the family), are notable for their complete regularity and, in special cases, complete transitivity (Gillespie et al., 2012). Complete regularity means that the code and its shells under the Hamming metric form equitable partitions, with well-defined intersection numbers. Complete transitivity is more restrictive, requiring group actions by automorphisms to be transitive within each shell. The Nordstrom–Robinson code, for $(m,\delta)=(16,6)$, is both completely regular and transitive, but general Preparata codes are completely regular without being transitive. This explicitly resolves the question of whether complete regularity always implies complete transitivity, revealing the distinctive symmetry profile among Preparata-like codes.

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Preparata-like codes thus occupy a central position in modern coding theory and finite geometry, providing links between algebraic code constructions, geometric partitioning, and advanced quantum coding strategies. Their classification hinges on metric rigidity, combinatorial design, and unique symmetry properties; their practical utility is bolstered by algorithmic advances in decoding exploiting their hidden linearity over rings. These codes continue to be a source of new mathematical constructions, insights into quantum error correction, and applications in communications and cryptography.