Non-Central Codes with Unit Constructions

• Non-Central Codes Using Units are codes derived from non-central units in group rings and division algebras, generating structures that extend beyond classical ideal-based approaches.
• They employ group ring-matrix isomorphisms to construct full-rank generator and check matrices, exemplified by codes like the [14,6,6] code over F₃.
• These constructions enable enhanced minimum distance and unique automorphism properties, though they present computational challenges in large algebraic settings.

Non-central codes using units arise from constructive methods in ring and group ring theory, exploiting the existence of units that do not lie in the center of the associated algebraic structure. This class of codes extends classical algebraic coding theory beyond ideal-based constructions, allowing generator and code structures fundamentally incompatible with two-sided ideal properties. Such codes display distinct properties, including asymmetry, non-equivalence to abelian codes, and, in the non-linear context, the ability to surpass traditional linear code parameters.

1. Algebraic Preliminaries: Units, Non-Centrality, and Group Rings

Let $G$ be a finite group of order $n$, and $R$ a (not necessarily commutative) ring with identity. The group ring $R[G]$ consists of finite formal sums $u = \sum_{g\in G}\alpha_g g$ with $\alpha_g\in R$, equipped with component-wise addition and convolution multiplication. Units $U(R[G])$ are those $u$ for which there exists $v\in R[G]$ with $uv=1=vu$.

Central units lie in the center

$$Z(R[G]) = \{z \in R[G] \mid zg = gz, \forall g\in G\}.$$

A unit $u\not\in Z(R[G])$ is non-central. Classical (ideal-based) group code theory is largely concerned with codes produced by central idempotents or units, for which group ring right (or left) ideals yield codes with group-invariant properties. Non-central units generate left or right modules that fail to be two-sided ideals, thereby expanding the structural diversity of available codes (0710.5893, Chahal et al., 5 Nov 2025).

2. Code Construction via Non-Central Units

A fundamental advance is the recognition that for any unit $u\in R[G]$, one can construct an associated $n\times n$ matrix via the group ring-matrix isomorphism $\varphi: R[G]\to M_n(R)$, defined with respect to a chosen ordering $G=\{g_1,\dots,g_n\}$. For $u = \sum_{i=1}^n \alpha_{g_i} g_i$,

$(\varphi(u))_{i,j} = \alpha_{g_i^{-1}g_j}.$

Given any selection $S=\{g_{j_1},...,g_{j_r}\}$, the submatrix $A$ consisting of the corresponding $r$ rows of $\varphi(u)$ forms a full-rank generator matrix of a linear $[n,r]$ code over $R$. The inverse $u^{-1}$ gives rise to the check matrix $H = D^T$, where $D$ is the submatrix of $\varphi(u^{-1})$ corresponding to the complement of $S$. This approach generalizes to codes over rings, matrix rings, or group rings themselves (0710.5893).

Non-central units produce codes $C=Wu$ for $W$ a submodule of $R[G]$ of rank $r<n$, which are not ideals: the non-commutativity $g u\neq u g$ for some $g$ ensures failure of two-sided closure unless $u$ is central.

3. Non-Central Codes Beyond Ideals: Properties and Concrete Constructions

Non-central codes admit wide latitude in structure. In metacyclic group algebras, central primitive idempotents $e$ derived from strong Shoda pairs $(H,K)$ generate classical two-sided central codes. To yield non-central codes, the idempotent is "cut" by a projector $\widehat{N}$ for a subgroup $N$, yielding $e^\beta = e\widehat{N}$, and then conjugated by a unit $u$ (often a Bass, bicyclic, or alternating unit): $e^{\beta,u}=u^{-1}e^\beta u$ (Chahal et al., 5 Nov 2025). The left ideal $I_u=F[G]\cdot e^{\beta,u}$ maintains dimension but can achieve strictly increased minimum weight compared to its central counterpart.

For example, in $F_3 D_{14}$ with $$D_{14}=\langle a,b\mid a^7=b^2=1,\, bab^{-1}=a^{-1}\rangle$$, the code $F_3 D_{14} \cdot f'$ (where $f' = u^{-1}f u$, $f=(1-\widehat{\langle a\rangle})(1+b)/2$ and $u=1+a$) is a non-central $[14,6,6]$ code, achieving the best known minimum distance for these parameters. This structural innovation is inaccessible to codes from abelian group rings, which display more rigid automorphism and weight structures. Non-central codes constructed this way are systematically nonequivalent to any abelian group code of the same length and dimension (Chahal et al., 5 Nov 2025).

4. Sum-Rank, Non-commutative Codes, and Division Algebra Units

In the context of sum-rank metric codes, non-centrality becomes essential when constructing codes from the norm-one units $O^1$ in a maximal order $O$ of a central division algebra $D/F$. The code map

$$\Theta: U = O^1 \to \prod_{v\in S} M_d(k_v) \cong M_d(k_0)^s$$

where $S$ is a set of split finite places and $k_0 \cong \mathbb{F}_{q_0}$, produces codewords as tuples of matrices, with the sum-rank distance

$d_R(x,y) = \sum_{v\in S} \text{rk}(x_v - y_v).$

A key result is the existence of asymptotically good families of such codes: for each $d\ge 2$, one obtains codes over block length $N=ds$ and alphabet $M_d(\mathbb{F}_p)$, with rate $R>0$, relative sum-rank distance $\delta>0$, and $\ln p=O(\ln d)$ (Maire et al., 2018). The non-commutativity of $O^1$ precludes centrality, and these codes extend the geometric philosophy—previously realized through commutative function field constructions—into fully non-commutative arithmetic settings.

5. Non-Central Codes from Units in Nonlinear and Convolutional Contexts

Non-central unit constructions are not restricted to linear codes or finite fields. Codes can be built as cosets in the group of units $U(R)$ of rings such as $\mathbb{Z}_4[G]$ for finite abelian $G$. For example, Best's $(10,40,4)$ code is realized as a coset $C = u H$ where $H\leq U(\mathbb{Z}_4[\mathbb{Z}_5])$ is an explicitly constructed subgroup and $u\in U(R)$. Under the Gray map, this produces binary nonlinear codes with minimum Lee distance corresponding to the Hamming metric in the image. Differences between codewords remain in $H H^{-1}$, ensuring structural parity for minimum distance analysis. Decoding exploits the group structure: invert, multiply by $u^{-1}$, and perform subgroup membership checks, yielding highly efficient syndrome algorithms (Greferath et al., 2011).

Extensions to convolutional codes are achieved by considering units in Laurent series over non-commutative matrix or group rings, $R[z, z^{-1}]$. With $u(z)=\sum U_i z^i$ a unit with inverse $v(z)$, algebraic block decompositions yield generator and check matrices for convolutional codes with precisely controlled free distances. In this setting, the non-commutative structure enables flexible design of memory and weight by tuning the nilpotency and placement of the coefficients $U_i$ (0711.3629).

6. Parameters, Advantages, and Limitations

Non-central codes using units have flexible parameters. The code length is set by $|G|$ (or analogously, the total block length in division algebra settings), and the dimension by the size of the selected basis or the order of the associated subgroup. Minimum distance estimates in non-central codes require combinatorial or group-theoretic analysis, sometimes via evaluation in group rings or via explicit calculation of idempotent supports after conjugation.

A salient advantage is that non-centrality opens up code parameter spaces inaccessible to ideal-based (central) constructions. Notably, best-known codes such as $[14,6,6]$ over $F_3$ and $[1000,r]$ LDPC codes over $F_2[C_{1000}]$ emerge, as do nonlinear codes with sizes exceeding the best linear competitors (Greferath et al., 2011). The algebraic structure allows tailoring codes to be LDPC, self-dual, or have exotic automorphism groups.

Potential limitations arise in the explicit construction of suitable non-central units with desired invertibility and support properties. Decoding and verification of minimum distance may require deeper utilization of the underlying non-commutative algebraic structure. For large underlying groups or rings, matrix manipulations can become computationally intensive, though practical implementations often only require certain submatrices for encoding (0710.5893).

7. Relation to Classical and Modern Coding Theory

The use of non-central units constitutes a significant generalization of group-code and ring-code paradigms. Whereas classical codes (e.g., cyclic, Reed–Solomon, BCH) arise from ideals and commutative settings, non-central constructions enlarge the universe of possible codes: one-sided modules, non-abelian symmetries, and non-linear Gray-lifted codes (0710.5893, Chahal et al., 5 Nov 2025, Greferath et al., 2011).

The algebraic mechanisms—particularly group ring-matrix isomorphisms, conjugation by units, and the exploitation of division algebraic and arithmetic group structures—demonstrate the fruitful interaction between advanced algebra and coding theory. These frameworks underpin ongoing developments in module-theoretic code families, arithmetic lattice codes, and sum-rank metric codes, and they continue to deliver codes matching or exceeding the best known bounds for both linear and nonlinear casework.