Function-Correcting Lee Codes (FCLCs)

• FCLCs are Lee-metric codes that ensure the reliable recovery of a function f(u) rather than the full message, offering reduced redundancy.
• They leverage irregular Lee-distance codes to establish Plotkin- and Gilbert–Varshamov-type bounds, linking redundancy to minimum required code length.
• Explicit constructions target canonical functions like Lee weight, weight distribution, and modular sum, tailoring redundancy based on both function properties and local constraints.

Function-Correcting Lee Codes (FCLCs) are systematic function-correcting codes over $\mathbb{Z}_m$ designed for Lee-metric channels: instead of requiring reliable recovery of the entire message, they require reliable recovery of a prescribed function $f(u)$ of the message $u$, and they do so by enforcing distance constraints only between codewords associated with different function values. In the current literature, this perspective yields an exact redundancy characterization through irregular Lee-distance codes, general Plotkin- and Gilbert–Varshamov-type bounds, exact or near-exact results for several structured function classes, and a direct specialization of homogeneous-distance FCCs over $\mathbb{Z}_4$, where homogeneous distance coincides with Lee distance (Verma et al., 23 Jul 2025).

1. Lee-metric setting and correction criterion

The ambient alphabet is $\mathbb{Z}_m$, $m\ge 2$, with Lee weight

$w_L(x)=\min\{x,m-x\},\qquad x\in\mathbb{Z}_m.$

Under the symmetric labeling of residues, this becomes $w_L(x)=|x|$. For vectors $u,v\in\mathbb{Z}_m^k$, the Lee distance is

$d_L(u,v)=\sum_{i=1}^k w_L(u_i-v_i).$

A Lee ball of radius $f(u)$0 around $f(u)$1 is $f(u)$2, and for $f(u)$3 its volume in $f(u)$4 is

$f(u)$5

which is the volume term used in Gilbert–Varshamov-type arguments (Verma et al., 23 Jul 2025).

An FCLC uses a systematic encoder

$f(u)$6

where $f(u)$7 is the redundancy. For a fixed function $f(u)$8 and Lee error radius $f(u)$9, the defining requirement is

$u$0

This is strictly weaker than a classical $u$1-error-correcting Lee code, which would require the same inequality for all $u$2. Accordingly, any conventional Lee code with minimum distance $u$3 is automatically an $u$4-FCLC for every $u$5, but usually with non-optimal redundancy (Verma et al., 23 Jul 2025).

The optimal redundancy is

$u$6

The decoding implication is purely metric: if the received word $u$7 satisfies $u$8, then all codewords within Lee radius $u$9 of $\mathbb{Z}_4$0 must correspond to the same function value, even if they do not determine the same message. A common misconception is that an FCLC is a weakened decoder for a conventional Lee code; in fact, the code design problem itself is different, because only function-distinct pairs must be separated (Verma et al., 23 Jul 2025).

2. Irregular Lee-distance codes and exact redundancy characterization

The central structural device is the irregular Lee-distance code. Given a symmetric nonnegative matrix $\mathbb{Z}_4$1, a set $\mathbb{Z}_4$2 is a $\mathbb{Z}_4$3-code in the Lee metric if

$\mathbb{Z}_4$4

The minimum length of such a code is denoted $\mathbb{Z}_4$5. When all off-diagonal entries of $\mathbb{Z}_4$6 equal a constant $\mathbb{Z}_4$7, this reduces to the standard parameter $\mathbb{Z}_4$8 for an $\mathbb{Z}_4$9-Lee code (Verma et al., 23 Jul 2025).

For messages $\mathbb{Z}_m$0, the distance requirement induced by $\mathbb{Z}_m$1 and $\mathbb{Z}_m$2 is captured by

$\mathbb{Z}_m$3

where $\mathbb{Z}_m$4. This matrix measures exactly how much Lee distance the redundancy part must contribute after the systematic part contributes $\mathbb{Z}_m$5 (Verma et al., 23 Jul 2025).

The exact equivalence theorem states that if $\mathbb{Z}_m$6, then

$\mathbb{Z}_m$7

Thus the optimal redundancy problem for FCLCs is equivalent to the shortest-length problem for an irregular Lee-distance code. A direct lower-bound corollary is that for any subset $\mathbb{Z}_m$8,

$\mathbb{Z}_m$9

This reduction is the Lee-metric analogue of the irregular-distance viewpoint used earlier for Hamming-metric FCCs and later for symbol-pair and sum-rank metrics (Verma et al., 23 Jul 2025, Xia et al., 2023, Kammila et al., 4 Jul 2026).

A second, smaller matrix acts on the image of $m\ge 2$0. If $m\ge 2$1, define

$m\ge 2$2

and then

$m\ge 2$3

This yields the general upper bound

$m\ge 2$4

obtained by assigning the same redundancy vector to all messages with the same function value. When one can choose representatives $m\ge 2$5 realizing the same matrix, the upper and lower bounds coincide and give an exact formula (Verma et al., 23 Jul 2025).

3. General redundancy bounds

The general lower bound is a Plotkin-type inequality for irregular Lee-distance codes. If $m\ge 2$6 is an $m\ge 2$7 Lee distance matrix, then

$m\ge 2$8

For regular distance $m\ge 2$9, this yields

$w_L(x)=\min\{x,m-x\},\qquad x\in\mathbb{Z}_m.$0

Two specializations are especially significant: for $w_L(x)=\min\{x,m-x\},\qquad x\in\mathbb{Z}_m.$1, Lee distance is Hamming distance, and the bound recovers the Hamming irregular-distance bound of Lenz et al.; for $w_L(x)=\min\{x,m-x\},\qquad x\in\mathbb{Z}_m.$2, where Lee distance coincides with homogeneous distance, the resulting bound improves the earlier $w_L(x)=\min\{x,m-x\},\qquad x\in\mathbb{Z}_m.$3 homogeneous-metric Plotkin bound of Liu–Liu by a factor of $w_L(x)=\min\{x,m-x\},\qquad x\in\mathbb{Z}_m.$4 (Verma et al., 23 Jul 2025).

The general upper bound is Gilbert–Varshamov-type: $w_L(x)=\min\{x,m-x\},\qquad x\in\mathbb{Z}_m.$5 It is obtained by a greedy construction that successively avoids Lee balls of the forbidden radii around previously chosen codewords (Verma et al., 23 Jul 2025).

A coarse but function-universal sandwich bound depends only on $w_L(x)=\min\{x,m-x\},\qquad x\in\mathbb{Z}_m.$6: $w_L(x)=\min\{x,m-x\},\qquad x\in\mathbb{Z}_m.$7 The lower bound comes from the existence, for every non-constant $w_L(x)=\min\{x,m-x\},\qquad x\in\mathbb{Z}_m.$8, of two messages at Lee distance $w_L(x)=\min\{x,m-x\},\qquad x\in\mathbb{Z}_m.$9 with different function values; the upper bound comes from assigning one codeword of an $w_L(x)=|x|$0-Lee code to each function value (Verma et al., 23 Jul 2025). The two-code case is exact: $w_L(x)=|x|$1 Hence every non-constant FCLC satisfies

$w_L(x)=|x|$2

For $w_L(x)=|x|$3 this becomes $w_L(x)=|x|$4, and for $w_L(x)=|x|$5 it becomes $w_L(x)=|x|$6 (Verma et al., 23 Jul 2025).

4. Locality in the Lee metric

A major structured subclass is defined through Lee-locality. For a function $w_L(x)=|x|$7, the function ball is

$w_L(x)=|x|$8

The function is locally $w_L(x)=|x|$9 if

$u,v\in\mathbb{Z}_m^k$0

If $u,v\in\mathbb{Z}_m^k$1, one obtains locally binary Lee functions (Verma et al., 23 Jul 2025).

The key combinatorial device is a coloring map $u,v\in\mathbb{Z}_m^k$2 such that

$u,v\in\mathbb{Z}_m^k$3

whenever $u,v\in\mathbb{Z}_m^k$4 or $u,v\in\mathbb{Z}_m^k$5. Using a length-$u,v\in\mathbb{Z}_m^k$6 Lee code $u,v\in\mathbb{Z}_m^k$7, one then encodes

$u,v\in\mathbb{Z}_m^k$8

which yields the general locality bound

$u,v\in\mathbb{Z}_m^k$9

for every locally $d_L(u,v)=\sum_{i=1}^k w_L(u_i-v_i).$0 function (Verma et al., 23 Jul 2025).

The locally binary case is completely characterized: $d_L(u,v)=\sum_{i=1}^k w_L(u_i-v_i).$1 whenever $d_L(u,v)=\sum_{i=1}^k w_L(u_i-v_i).$2 is locally $d_L(u,v)=\sum_{i=1}^k w_L(u_i-v_i).$3. This gives an exact redundancy theorem rather than a one-sided bound (Verma et al., 23 Jul 2025).

Later explicit-construction work over $d_L(u,v)=\sum_{i=1}^k w_L(u_i-v_i).$4 sharpened the locality picture for locally $d_L(u,v)=\sum_{i=1}^k w_L(u_i-v_i).$5-bounded functions. If $d_L(u,v)=\sum_{i=1}^k w_L(u_i-v_i).$6, it gives a direct scalar-parity construction with

$d_L(u,v)=\sum_{i=1}^k w_L(u_i-v_i).$7

obtained by spacing $d_L(u,v)=\sum_{i=1}^k w_L(u_i-v_i).$8 parity symbols in Lee distance and repeating the chosen symbol $d_L(u,v)=\sum_{i=1}^k w_L(u_i-v_i).$9 times (K. et al., 3 Aug 2025).

5. Canonical function classes

Three function classes dominate the FCLC literature: Lee weight, Lee weight distribution, and modular sum. Over $f(u)$00, the same classes also appear in the homogeneous-distance FCC literature because homogeneous distance and Lee distance coincide there (Liu et al., 4 Jul 2025).

Function class	Stated Lee-metric result	Source
$f(u)$01	$f(u)$02	(Verma et al., 23 Jul 2025)
$f(u)$03	$f(u)$04	(Verma et al., 23 Jul 2025)
$f(u)$05	explicit FCLCs; $f(u)$06 for odd $f(u)$07 with $f(u)$08, and $f(u)$09 for even $f(u)$10	(K. et al., 3 Aug 2025)
locally $f(u)$11-bounded $f(u)$12	$f(u)$13 when $f(u)$14	(K. et al., 3 Aug 2025)

For the Lee weight function $f(u)$15, the image is $f(u)$16. The triangle inequality yields

$f(u)$17

for every $f(u)$18, so $f(u)$19 is locally $f(u)$20. At $f(u)$21, this gives

$f(u)$22

A stronger result uses explicit representatives $f(u)$23 satisfying

$f(u)$24

Consequently,

$f(u)$25

and

$f(u)$26

For $f(u)$27, one obtains the explicit lower bound

$f(u)$28

which over $f(u)$29 specializes to

$f(u)$30

That $f(u)$31 specialization exactly matches the earlier Liu–Liu lower bound, but the Lee-metric derivation holds for arbitrary $f(u)$32 (Verma et al., 23 Jul 2025).

For the Lee weight distribution function

$f(u)$33

the same weight-difference estimate implies

$f(u)$34

Hence $f(u)$35 is locally $f(u)$36, and at $f(u)$37,

$f(u)$38

This makes the dependence on the quantization parameter $f(u)$39 explicit: coarser quantization lowers the effective local alphabet size and can reduce redundancy (Verma et al., 23 Jul 2025).

The modular sum function was treated explicitly in later Lee-metric construction work. Over $f(u)$40, if

$f(u)$41

then there are explicit parity-symbol constructions with redundancy $f(u)$42 for odd $f(u)$43 and $f(u)$44, and redundancy $f(u)$45 for even $f(u)$46. The same work also derived lower bounds from a refined Plotkin-like bound and proved optimality in several concrete parameter regimes (K. et al., 3 Aug 2025).

Over $f(u)$47, the homogeneous-distance study gives additional exact small-$f(u)$48 statements because Lee and homogeneous metrics coincide there. For the Lee weight function on $f(u)$49, it yields

$f(u)$50

and supplies explicit constructions achieving those values (Liu et al., 4 Jul 2025).

6. Relation to adjacent FCC frameworks

FCLCs belong to a broader FCC program in which the metric changes but the function-centric logic remains stable. The Hamming-metric literature established the general FCC paradigm and proved the universal lower bound $f(u)$51 over finite fields, with equality for sufficiently large fields; symbol-pair and sum-rank works then recast the same design problem as one of irregular-distance coding in their respective metrics (Ly et al., 19 Apr 2025, Xia et al., 2023, Kammila et al., 4 Jul 2026). This suggests that the Lee-metric theory is not an isolated construction but one instance of a metric-dependent, irregular-distance framework.

At the same time, the Lee case has distinctive algebraic features. On $f(u)$52, the homogeneous weight

$f(u)$53

is exactly the Lee weight, so every FCCHD over $f(u)$54 is literally an FCLC (Liu et al., 4 Jul 2025). For $f(u)$55 with $f(u)$56, homogeneous distance remains a closely related but broader framework, and later work on finite chain rings extends locality, linear-function bounds, and explicit constructions in that setting (Verma et al., 15 Mar 2026).

A further branch studies strict FCCs with data protection, where the function is protected at a higher distance level than the underlying data. Those results are currently stated in the Hamming metric and use $f(u)$57-distance graphs, Cayley-graph structure, and subcodes generated by low-weight codewords (Rajput et al., 1 Mar 2026, Rajput et al., 29 Apr 2026). A plausible implication is that analogous Lee-distance graphs and Lee-weight-generated subcodes could support a strict Lee-metric theory, but that step is presented as a prospective direction rather than an established Lee-metric theorem in the cited works.

Across this literature, the central distinction remains the same: FCLCs are not merely Lee-metric error-correcting codes with a specialized decoder; they are codes whose distance constraints, redundancy bounds, and explicit constructions are tailored to the function itself. Later explicit comparisons with classical Lee error-correcting codes and codes that encode only function values show that FCLCs can significantly reduce redundancy while preserving function correctness (K. et al., 3 Aug 2025).