--- title: Irregular Lee Distance Codes url: https://www.emergentmind.com/topics/irregular-lee-distance-codes type: topic --- # Irregular Lee Distance Codes Irregular Lee distance codes are code families in Lee-metric spaces for which uniform pairwise separation, perfect tiling structure, or homogeneous algebraic behavior is relaxed or absent. In the cited literature, the phrase does not denote a single universally fixed class. It can refer to arbitrary codes \(C\subseteq \mathbb{Z}_q^n\) with no linearity or regularity assumption, to explicitly defined \(\mathbf{D}\)-irregular-Lee-distance codes whose pairwise lower bounds are prescribed by a matrix, and to non-perfect, non-periodic, non-free, or heterogeneous Lee-metric constructions that arise when perfect Lee sphere tilings fail or when coordinate behavior varies across a ring-filtration or weight-spectrum hierarchy [1810.05066], [2508.01702], [1701.08412]. ## 1. Formal setting and meanings of irregularity The Lee metric on \((\mathbb{Z}/q\mathbb{Z})^n\) is \[ d_L(x,y)=\sum_{i=1}^{n}\min\{|x_i-y_i|,\;q-|x_i-y_i|\}, \] and on \(\mathbb{Z}^n\) it becomes the \(L^1\) metric \[ d_L(x,y)=\sum_{i=1}^n |x_i-y_i|. \] A code in the Lee metric is any subset \(C\subseteq \mathbb{Z}_q^n\), and no linearity or regularity is assumed in the general definition. The extremal size parameter is \[ A_q^L(n,d)=\max\{|C|: C\subseteq \mathbb{Z}_q^n,\ d_{\min}(C)\ge d\}, \] with \(d_{\min}(C)\) the minimum Lee distance of \(C\) [1810.05066]. Several distinct notions of “irregularity” appear in the literature. | Sense | Formal object | Source | |---|---|---| | Arbitrary Lee code | Any \(C\subseteq \mathbb{Z}_q^n\) | [1810.05066] | | Distance-matrix irregularity | \(\mathbf{D}_L\)-code with \(d_L(\mathbf{p}_i,\mathbf{p}_j)\ge [\mathbf{D}]_{ij}\) | [2508.01702] | | Non-perfect/non-tiling behavior | Codes without perfect Lee sphere tilings | [1701.08412], [1804.09290] | | Ring-theoretic heterogeneity | Non-free codes, support subtypes, filtration layers | [2112.06635], [2307.06079] | | Weight-spectrum irregularity | Many distinct Lee weights or non-few-weight behavior over \(\mathbb{Z}_4\) | [2502.10697] | The explicit distance-matrix notion is the most formal. A set \(\mathcal{P}=\{\mathbf{p}_1,\dots,\mathbf{p}_M\}\subseteq \mathbb{Z}_q^r\) is a \(\mathbf{D}\)-irregular-Lee-distance code if there exists an ordering such that \[ d_L(\mathbf{p}_i,\mathbf{p}_j)\ge [\mathbf{D}]_{ij}\qquad \forall i,j\in[M], \] and the minimum feasible length is denoted \(N_L(\mathbf{D})\) [2508.01702]. When \([\mathbf{D}]_{ij}=D\) for all \(i\neq j\), this reduces to the usual regular-distance shortest-length problem \(N_L(M,D)\). A different but complementary use of “irregular” arises in \(\mathbb{Z}_4\)-linear constructions. There the focus is not on pair-specific minimum distances but on non-uniform or highly constrained Lee weight distributions. For odd \(m\), some defining-set constructions produce 3-weight or otherwise controlled multi-weight Lee codes, whereas for even \(m\) the same constructions produce many more distinct Lee weights and are described as “performing worse,” with full weight distributions left undetermined [2502.10697]. ## 2. Perfect tilings, nonexistence, and why irregularity becomes necessary Perfect Lee codes are the regular benchmark against which irregularity is often measured. In \((\mathbb{Z}/q\mathbb{Z})^n\), a perfect \(e\)-error-correcting Lee code is a subset \(C\) such that every vector lies within Lee distance at most \(e\) from a unique codeword. In \(\mathbb{Z}^n\), this is equivalent to a translational tiling by Lee spheres \(S(n,e)\). For \(q\ge 2e+1\), there is a natural bijection between \(PL(n,e,q)\)-codes and \(PL(n,e)\)-codes that are unions of cosets of \(q\mathbb{Z}^n\), so perfect Lee codes are exactly translational tilings of \(\mathbb{Z}^n\) by Lee spheres, up to periodicity mod \(q\) [1701.08412]. The central obstruction is the Golomb–Welch conjecture: for \(n\ge 3\) and \(e\ge 2\), there exist no \(PL(n,e)\)-codes. For \(e=2\), the Lee sphere size is \[ |S(n,2)|=2n^2+2n+1. \] A number-theoretic nonexistence theorem proves that if \(p=2n^2+2n+1\) is prime, \(a\) is the least positive integer with \(p\mid 4^a+4n+2\), \(b\) is the least positive integer with \(p\mid 4^b-1\), and the Diophantine equation \[ a(x+1)+by=n,\qquad x,y\in\mathbb{Z}_{\ge 0} \] has no solution, then no \(PL(n,2)\)-code exists. This excludes, for example, \[ n=5,7,9,12,14,17,\ldots \] and is expected to apply to infinitely many dimensions if there are infinitely many primes of the form \(2n^2+2n+1\) [1701.08412]. For linear perfect 2-error-correcting Lee codes, the obstruction is stronger. Let \[ \mathcal{N}=\{n\ge 3:\text{there are no linear }PL(n,2)\text{-codes}\}. \] One result shows that \[ n\equiv 8,13,18,23 \pmod{25} \] implies \(\mathrm{LPL}(n,2)=\emptyset\), hence \[ \underline{d}(\mathcal{N})\ge \frac{4}{25} \quad\text{and}\quad \#\{n\in\mathcal{N}:n\le x\}\ge \frac{4x}{25}(1+o(1)). \] Thus a positive lower density of dimensions forbid linear perfect 2-error-correcting Lee codes [1804.09290]. A further algebraic treatment via group rings and characters proves nonexistence of linear perfect Lee codes of radii \(2\) and \(3\) for infinitely many dimensions, and for radius \(2\) it excludes all \(3\le n\le 100\) except \[ n=16,21,36,55,64,66,78,92. \] This is obtained by translating the perfect tiling condition into group-ring identities such as \[ T^2=2G-T^{(2)}+2n \] for radius \(2\), then deriving contradictions via character values, finite-field reductions, and arithmetic conditions on prime divisors of \(2n^2+2n+1\) [1802.04608]. These results do not directly address nonlinear perfect Lee codes. In particular, the positive-density nonexistence argument for linear \(PL(n,2)\)-codes uses quotient groups and homomorphisms, and there is no immediate analogue for a general nonlinear code [1804.09290]. This clarifies a common misconception: linear nonexistence strengthens the Golomb–Welch picture but does not settle the nonlinear case. Still, the cumulative effect is clear. In many parameter regimes, perfect regular tilings are impossible, so any high-performance Lee code must be non-perfect, non-lattice, non-periodic, or otherwise irregular. ## 3. Distance-matrix irregularity and function-correcting Lee codes The most explicit modern formalization of irregular Lee-distance coding appears in the theory of function-correcting Lee codes (FCLCs). Here one does not need uniform separation between all message pairs. One only needs enough separation to recover a function value \(f(\mathbf{u})\) from a Lee-corrupted systematic encoding \(\mathrm{Enc}(\mathbf{u})=(\mathbf{u},p(\mathbf{u}))\) when at most \(t\) Lee errors occur [2508.01702]. For a function \(f:\mathbb{Z}_q^k\to \mathrm{Im}(f)\), the distance requirement matrix on a message set \(\{\mathbf{u}_1,\dots,\mathbf{u}_M\}\) is \[ [\mathbf{D}_f(t,\mathbf{u}_1,\dots,\mathbf{u}_M)]_{ij} = \begin{cases} [2t+1-d_L(\mathbf{u}_i,\mathbf{u}_j)]^+,& f(\mathbf{u}_i)\neq f(\mathbf{u}_j),\\ 0,& \text{otherwise}, \end{cases} \] where \([x]^+=\max\{x,0\}\). The optimal redundancy is exactly the shortest length of an irregular Lee-distance code with this matrix: \[ r_L^f(q,k,t)=N_L\big(\mathbf{D}_f(t,\mathbf{u}_1,\dots,\mathbf{u}_{q^k})\big). \] This identity shows that irregular Lee-distance coding is not merely a relaxation of classical minimum-distance coding; it is the natural geometry of function protection under Lee-metric channels [2508.01702]. A reduced description replaces the full \(q^k\times q^k\) matrix by a function-distance matrix on \(\mathrm{Im}(f)\). For function values \(f_1,f_2\), \[ d_L^f(f_1,f_2) = \min_{\mathbf{u}_1,\mathbf{u}_2} d_L(\mathbf{u}_1,\mathbf{u}_2) \quad \text{s.t. } f(\mathbf{u}_1)=f_1,\ f(\mathbf{u}_2)=f_2, \] and then \[ [\mathbf{D}_f(t,f_1,\dots,f_E)]_{ij} = \begin{cases} [2t+1-d_L^f(f_i,f_j)]^+,& i\neq j,\\ 0,& i=j. \end{cases} \] This yields \[ r_L^f(q,k,t)\le N_L(\mathbf{D}_f(t,f_1,\dots,f_E)), \] with equality when representatives realizing all pairwise function distances exist [2508.01702]. The paper also proves a Plotkin-like lower bound for irregular Lee-distance codes. If \(\mathbf{D}\in\mathbb{N}_0^{M\times M}\), then \(N_L(\mathbf{D})\) is bounded below by a multiple of \(\sum_{i1\), Lee-equidistant codes have rank at most \(2\), and the paper completes Wood’s classification by giving explicit minimal-length constructions for rank \(1\) and rank \(2\) cases [2112.06635]. This is an important corrective to the intuition imported from field-based Hamming theory: extremal Lee behavior over rings is often non-free and highly torsion-sensitive. A different refinement of Lee irregularity appears in generalized Lee weights. Over \(Z=\mathbb{Z}/p^s\mathbb{Z}\), three support notions are introduced: support subtype \((n_0(C),\dots,n_s(C))\), join-Lee support, and column-Lee support. The support subtype records, coordinate by coordinate, which ideal \(\langle p^i\rangle\) is generated by the coordinate projection of the code; this already encodes heterogeneous coordinate activity. Join-Lee generalized weights \[ d_L^{\vee,r}(C) = \min\{|\supp_L^{\vee}(D)|:D\le C,\ \mathrm{rk}(D)=r\} \] reduce to generalized Hamming weights of the socle scaled by \(M_{s-1}\), giving the bound \[ d_L(C)\le M_{s-1}(n-K+1) = \Big\lfloor\frac{p}{2}\Big\rfloor p^{s-1}(n-K+1). \] Column-Lee generalized weights and filtration subcodes \[ C_i=C\cap \langle p^i\rangle^n \] yield stronger Singleton-like bounds that explicitly track where rows and columns vanish along the \(p\)-adic filtration [2307.06079]. The filtration viewpoint is especially relevant to irregular Lee codes because it captures multi-level protection and heterogeneous coordinates directly. Instead of measuring only the rank of a subcode, it measures how deep subcodes lie inside the ideal filtration. This provides upper bounds on \(d_L(C)\) that are sensitive to the detailed support profile of the code rather than just its ambient length and size [2307.06079]. Over \(\mathbb{Z}_4\), a complementary sort of irregularity appears in Lee weight distributions. Using trace-evaluation codes over \(\mathrm{GR}(4,m)\), several classes of linear codes have explicitly determined Lee weight distributions for odd \(m\). For \(D=D_{t_1}\cup D_{t_2}\) with \(t_1,t_2\) of the same parity, one obtains 3-weight Lee codes of length \(2^{m-1}\). For \(D=D_t\) and for complements \(L\setminus D_t\), one obtains structured multi-weight codes with weights of the form \(2^{m-2}\) or \(3\cdot 2^{m-2}\) perturbed by terms of order \(2^{(m-1)/2}\). For even \(m\), the same constructions yield many more distinct Lee weights and are described as “performing worse,” with full weight distributions not determined [2502.10697]. Here irregularity is spectral rather than geometric: the code remains linear, but its Lee weight spectrum becomes much less uniform. ## 6. Constructions, long-code asymptotics, and open directions The absence of perfect regular Lee codes does not imply a lack of strong constructive families. One route is via long algebraic-geometric constructions. Concatenation of an outer AG code over \(GF(Q)\) with an inner short Lee-metric code over \(\mathbb{Z}_q\) yields families whose rate \(R\) and relative Lee distance \(\delta\) satisfy \[ \frac{R}{k/n}+\frac{\delta}{d_L/n}\ge 1-\frac{1}{q^{k/2}-1}, \] where \([n,k,d_L]\) are the inner parameters. With Lee-metric BCH inner codes over \(\mathbb{Z}_p\), the resulting asymptotic line is approximately \(2R+\delta\approx 1\) for suitable \(p\) and \(t\) [1302.2246]. A more intrinsic AG bound, the corrected “Victorian” bound, shows that from a curve of genus \(g\) over \(\mathbb{F}_q\) with at least \(n+1\) rational points one obtains shortened codes over \(\mathbb{F}_q\) with parameters \([n-1,r-g]\) and Lee distance \[ d_L\ge \frac{n^2-r^2}{4r}, \] for \(2g-2