Function-Correcting Codes: Theory & Applications

• Function-Correcting Codes (FCCs) are error-correcting codes that ensure accurate recovery of specific function values by relaxing traditional distance constraints.
• They employ a combinatorial framework and metrics like the Lee metric to reduce redundancy, often achieving optimal performance with minimal extra bits.
• FCCs leverage irregular-distance coding and graph-theoretic techniques to enhance data aggregation and distributed computations in practical settings.

Function-Correcting Codes (FCCs) are a novel class of error-correcting codes designed to guarantee the reliable recovery of prescribed function values computed on a message, while minimizing the redundancy compared to classical codes. In contrast to traditional error-correcting codes (ECCs), which require all distinct message codewords to be separated by a fixed metric distance, FCCs only impose distance constraints between codewords with nonequivalent function values. This fundamental relaxation can dramatically reduce redundancy when the function of interest is many-to-one. FCCs admit a general, function- and metric-dependent combinatorial framework, and have been explicitly studied for various algebraic and combinatorial metrics, including Hamming, Lee, pair, and homogeneous distances.

1. Formal Definition and Framework

Consider messages $u \in \mathbb{Z}_q^k$ and a target function $f: \mathbb{Z}_q^k \rightarrow \operatorname{Im}(f)$. A systematic function-correcting code (FCC) for $f$ and $t$-error correction in a metric $d$ is an encoding map $$\operatorname{Enc}(u) = (u, p(u)) \in \mathbb{Z}_q^{k+r}$$ such that if $f(u_1) \neq f(u_2)$, then $$d(\operatorname{Enc}(u_1), \operatorname{Enc}(u_2)) \geq 2t+1$$ (K. et al., 3 Aug 2025). The smallest $r$ for which such an encoder exists is the optimal redundancy $r^f(q,k,t)$.

FCCs in the Lee Metric

The Lee metric for $a,b \in \mathbb{Z}_q$ is $d_L(a,b) = \min(|a-b|, q-|a-b|)$. For vectors, $d_L(x,y) = \sum_{i=1}^n d_L(x_i, y_i)$. The systematic FCC for $f$ in the Lee metric (a Function-Correcting Lee Code, FCLC) satisfies $$d_L(\operatorname{Enc}(u_1), \operatorname{Enc}(u_2)) \geq 2t+1$$ when $f(u_1) \neq f(u_2)$ (K. et al., 3 Aug 2025).

A central tool is the distance requirement matrix $D_f$ of size $M = |\mathbb{Z}_q^k|$ with entries $[D_f]_{ij} = [2t+1 - d_L(u_i, u_j)]^+$ for $f(u_i) \neq f(u_j)$, else $0$. The minimal $r$ such that a set of redundancy vectors $\{p_1, ..., p_M\} \subset \mathbb{Z}_q^r$ exists with $d_L(p_i, p_j) \geq [D_f]_{ij}$ for all $i \neq j$ is denoted $N_L(D_f)$. The optimal redundancy is $r_L^f(q,k,t) = N_L(D_f)$ (K. et al., 3 Aug 2025).

2. Core Principles and Theoretical Bounds

FCC design reduces to constructing irregular-distance codes: vector sets subject to a matrix of minimum distance constraints reflecting function preimages. Two main types of bounds result:

• Plotkin-like Lower Bound: For even $q$, $N_L(D) \geq (8/(M^2 q)) \sum_{i<j} [D]_{ij}$. For odd $q$, $$N_L(D) \geq (4/(M^2 \lfloor q/2 \rfloor)) (q/(q+1)) \sum_{i<j} [D]_{ij}$$. The denominator adjusts for parity in $M$ and $q$ (K. et al., 3 Aug 2025).
• Gilbert–Varshamov Upper Bound: $$N_L(D) \leq \min \{r : q^r > \max_j \sum_{i < j} V_L(r, D_{ij} - 1)\}$$, with $V_L(r, \rho)$ the size of a Lee ball of radius $\rho$ (Verma et al., 23 Jul 2025).

These generalize bounds for classical codes. For any function, the redundancy obeys

$$N_L(2, 2t) \leq r_L^f(q,k,t) \leq N_L(|\operatorname{Im}(f)|, 2t).$$

3. Explicit FCC Constructions and Function Classes

Explicit and often optimal FCLCs have been constructed for several function classes (K. et al., 3 Aug 2025, Verma et al., 23 Jul 2025):

3.1 Lee Weight ($f(u) = \mathrm{wt}_L(u)$)

Let $E = k \lfloor q/2 \rfloor + 1$; consider representative vectors of weights $0, 1, ..., E$. The code distance matrix is $[D_{wt}]_{ij} = [2t+1 - |i-j|]^+$.

Construction: For $q \geq 5$, $t \leq (q-3)/2$:

• Odd $q$: $p_s = 2s \bmod q$.
• Even $q$: $p_s = 2s \bmod q$ for $s < q/2$, $p_s = 2s+1 \bmod q$ for $s \geq q/2$.
• Redundancy is $t$ ($r = t$).

This is optimal for many small $q, k$. Applying the Plotkin bound to $D_{wt}$ gives explicit lower bounds (K. et al., 3 Aug 2025).

3.2 Lee-Weight Distribution ($f(u) = \lfloor \mathrm{wt}_L(u)/T \rfloor$)

Divide weight into $T$-bins. If $T$ divides $E$, then image size is $E' = E / T$.

• Upper bound: $r_L^\Delta(q,k,t) \leq N_L(D_\Delta)$, with $D_\Delta$ reflecting minimum bin differences.
• For $t \leq T$, redundancy $r \leq t$.

3.3 Modular Sum ($f(u) = \sum_{i} u_i \bmod q$)

Image size $E=q$. Assign each modular sum $s$ a parity $p_s$ as above. Redundancy requirements mirror those for Lee weight.

3.4 Locally $(2t,\lambda)_L$-Bounded Functions

If $f$ is locally $(2t,\lambda)_L$, i.e., the value set of $f$ within any Lee-ball of radius $2t$ is $\leq \lambda$, assign a coloring $\operatorname{Col}_f$ with $\lambda$ colors, then use a code with parameters adapted to $\lambda$. For $\lambda \leq q/2$, $r = \lceil t / \lfloor q/(2\lambda) \rfloor \rceil$. This is sometimes optimal by matching lower bounds.

4. Redundancy Analysis and Comparative Performance

Lower and upper bounds can often be explicitly calculated for concrete function classes using matrix methods and combinatorial arguments (K. et al., 3 Aug 2025). In the most studied cases, explicit constructions achieve minimal redundancy.

A comparative summary:

Method	Redundancy Lower Bound	Typical Redundancy Achieved
Classical Lee ECC	$\geq \log_q V_t^{(n)}$	Sphere-packing dominated
ECC on function values	$$\geq \log_q (|\operatorname{Im}(f)| \cdot V_t^{(n)})$$	Function-image size dominates
FCLC	$t, 2t-1$ (varies by class)	Often $r = t$ (see Lee weight)

In the Lee-weight example with $q=5,k=2,t=1$, ECC on data needs $r \geq 2$, ECC on function values $r \geq 3$, but FCLC achieves $r=1$. For large $k$ and functions with small image size, $r$ can be substantially smaller with FCLCs.

5. Graph-Theoretic and Algorithmic Perspectives

FCC construction is equivalent to finding large independent sets in a function-dependent graph where vertices encode messages and their redundancy and edges correspond to forbidden pairs under the metric and $f$ constraints (K. et al., 3 Aug 2025). The problem reduces to constructing a code with a prescribed irregular-distance matrix, leveraging coloring and code-assignment techniques. Greedy and coloring-based algorithms derive from Brooks' theorem and sphere-packing ideas.

6. Specializations, Metric Extensions, and Open Problems

The FCC paradigm generalizes readily to other metrics (b-symbol, symbol-pair, homogeneous, Hamming). Reed–Muller, Gray-code, and parity-repetition constructions have broad applicability (K. et al., 3 Aug 2025, Verma et al., 23 Jul 2025). Tight Plotkin-type bounds have been extended to the Lee metric, improving upon prior results in the homogeneous metric (Verma et al., 23 Jul 2025).

Open problems include:

• Tightening bounds for specific $f, t, q$.
• Explicit constructions for more general algebraic and combinatorial $f$.
• Extensions to list decoding, probabilistic decoding, and channels with additional error models.

7. Significance and Applications

FCCs, particularly in the Lee metric, provide a rigorous method to optimize and minimize redundancy when only a function of a message matters for downstream reliability—central to data aggregation, hash verification, machine learning inference in storage, and error-resilient distributed computations (K. et al., 3 Aug 2025). Their design paradigm offers a strict generalization of error-correcting codes, interpolating between function-value and data protection, and enables significant improvements in code rate for a wide spectrum of practical coding scenarios.