Metacyclic p-Group Codes

• Metacyclic p-group codes are linear codes formed as left ideals in group algebras of non-abelian metacyclic p-groups, defined by cyclic normal subgroups and structured idempotents.
• They are constructed through explicit Wedderburn decompositions and the enumeration of primitive central idempotents via strong Shoda pairs, ensuring clear parameter computation.
• Non-central codes, obtained by conjugating idempotents with specialized group units, often yield improved minimum distances and best known parameters in practical applications.

A metacyclic $p$-group code is a linear code realized as a left ideal in the group algebra $F_q[G]$, where $G$ is a finite non-abelian metacyclic $p$-group and $F_q$ is a finite field whose characteristic does not divide $|G|$. The paper of these codes intertwines the explicit Wedderburn decomposition of semisimple group algebras, the enumeration and structure of primitive central idempotents via strong Shoda pairs, and the construction of both central and non-central codes, including instances yielding best known parameters. These results draw together the algebraic theory of group algebras, the structure theory of metacyclic groups (notably dihedral and quaternion types), and explicit combinatorial constructions for code design.

1. Structural Foundations and Definitions

A group $G$ is called metacyclic if it possesses a cyclic normal subgroup $N$ such that $G/N$ is cyclic. In the context of $p$-groups, this typically yields presentations of the form

$$G \cong \langle a, b \mid a^{p^m} = 1,\, b^{p^\ell} = 1,\, b^{-1}ab = a^r \rangle$$

for suitable $m, \ell$, and $r$ of order $p^\ell$ modulo $p^m$. The group algebra $F_qG$ is semisimple when $\mathrm{char}(F_q) \nmid |G|$, and all linear codes of length $|G|$ over $F_q$ realized as left ideals are called group codes.

The Wedderburn structure theorem applies: $F_qG \cong \bigoplus_{i=1}^t M_{n_i}(D_i)$ where each $D_i$ is a finite field extension of $F_q$, and the decomposition is determined by the set of central primitive idempotents (pcis). Each code $C$ corresponding to a (central) pci $e$ has parameters $$[n = |G|,\, k = \dim_{F_q} eF_qG,\, d = \mathrm{wt}(C)]$$.

2. Strong Shoda Pairs and Wedderburn Decomposition

Strong Shoda pairs $(H, K)$ play a key role in producing all primitive central idempotents of $F_qG$. A pair $(H, K)$ with $K \triangleleft H \triangleleft N_G(K)$, $H/K$ cyclic and maximal abelian in $N_G(K)/K$, determines an explicit idempotent via

$$\epsilon(H, K) = |K|^{-1}\sum_{h \in H} \mathrm{tr}_{F_q(\xi_{|H/K|})/F_q}(\chi(hK)) h^{-1}$$

where $\chi$ is indexed by a $q$-cyclotomic orbit. The set of all $G$-conjugates of such elements provides mutually orthogonal pcis. Each block $F_qGe_C(G, H, K)$ is isomorphic to a matrix algebra over an appropriate field extension, with dimension and code parameters computed from the group indices and cyclotomic orbits.

In explicit families:

• For dihedral groups $D_{2^{n+1}}$ and quaternion groups $Q_{2^{n+1}}$, strong Shoda pairs can be listed combinatorially.
• In general odd $p$-group cases, the pairs reflect the maximal cyclic subgroups and their conjugates, with the resulting enumeration controlled by the possible q-cyclotomic orbits.

3. Explicit Idempotents and Minimal Codes

Explicit formulas for idempotents in familiar metacyclic $p$-groups are given in terms of trace maps in cyclotomic extensions. For $G = D_{2^{n+1}}$, the idempotents associated to $(\langle a \rangle, \langle a^{2^j}\rangle)$ are

$$e_{2^j, k} = 2^{-j} \widehat{\langle a^{2^j}\rangle} \sum_{i=0}^{2^j-1} \mathrm{tr}_{F_q(\zeta_{2^j})/F_q}(\zeta_{2^j}^{k i}) a^{-i}$$

and similar formulas exist in quaternion and general metacyclic cases, with possible refinements in block structure depending on the splitting of the relevant cyclotomic polynomials over $F_q$.

The minimal left ideals of $F_qG$, corresponding to central primitive idempotents, yield irreducible (minimal) codes. Further, if the matrix algebra block $M_r(F_{q^s})$ appears in the Wedderburn decomposition, it can be refined to $r$ non-central minimal left ideals, using primitive matrix units. These "cyclotomic" minimal codes are typically non-central and may not be permutation equivalent to any code from a group algebra of an abelian group.

4. Enumeration, Parameters, and Distance Bounds

The enumeration of central codes is directly accessible from the catalog of strong Shoda pairs and their associated cyclotomic orbit counts:

• For $D_{2^{n+1}}$, the number of pcis (and thus central minimal codes) is $4 + \sum_{j=2}^n \varphi(2^j)/o_{2^j}(q)$.
• For general metacyclic $p$-groups $G_{p^{n+1}}$, the number is $$1 + \varphi(p)/o_p(q) + \sum_{j=1}^{n-1} p \cdot \varphi(p^j)/o_{p^j}(q) + \varphi(p^n)/o_{p^n}(q)$$.

Dimensionality and distance formulas are as follows. For a cyclic normal subgroup $K \triangleleft G$ of order $m$ and the idempotent $e$ corresponding to $(G, G, K)$,

$\dim_{F_q} F_qG e = o_m(q)$

and the minimum distance satisfies

$2|K| \leq d \leq \mathrm{wt}(e)$

with equality in the lower bound for cases with $o_{p^j}(q) = \varphi(p^j)$. For blocks associated with $(\langle a \rangle, \langle a^{p^j} \rangle)$, the dimension multiplies by $[G:\langle a\rangle]$.

5. Non-Central Codes and Optimization via Group Units

Non-central codes are constructed by conjugating idempotents with units in $F_qG$, including Bass, bicyclic, or alternating units. When a block $F_qG e$ is isomorphic to $M_r(F_{q^s})$, left ideals corresponding to distinct matrix-units can give rise to inequivalent codes, as permutation equivalence does not relate these. Such non-central codes often achieve strictly larger minimum distances than central ones, with explicit improvements constructed for several groups and fields.

Examples:

• For $F_3D_{14}$, the central cyclotomic code $e_{7,1}$ yields $[14, 6, 4]$, while conjugation by a bicyclic unit boosts the distance to 6, producing a code inequivalent to any abelian code and attaining the best known parameters.
• For $F_5D_{14}$, a similar improvement yields a $[14, 6, 7]$ code.
• For metacyclic groups of order 39 over $F_2$, the minimum distance can be raised from 2 (central) to nearly the theoretical maximum through successive conjugations.

Such units are defined either as combinations of group elements (bicyclic units $1 + (1-h)g\hat{h}$), as sums over group generators (alternating units in characteristic 2), or through more sophisticated constructions (Bass units).

6. Illustrative Examples and Applications

Explicit construction in dihedral and quaternion cases showcases the full parameter lists of minimal codes:

• For $D_{2^{n+1}}$ over $F_q$ with $q$ odd, the family of codes includes $[2^n, 1, 2^n]$, $[2^{n+1}, o_{2^n}(q), 2]$, and further codes from matrix blocks with higher dimensionality and distance.
• The quaternion group $Q_{2^{n+1}}$ yields essentially the same code parameters, but in the non-split field case, matrix blocks introduce codes of twice the dimension but the same distance.
• Odd-$p$ metacyclic groups provide layered code structures, with top and intermediate blocks yielding codes of the form $[p^{n+1}, p o_{p^n}(q), 2p]$.

These explicit families furnish a broad spectrum of group code parameters, and the mechanisms for non-central code construction assure that metacyclic $p$-groups are sources of codes with best known or near-best minimum distances and strict inequivalence to abelian group codes.

7. Proof Techniques and Theoretical Underpinnings

The dimension and structure results rely on:

• Trace-based computations in cyclotomic field extensions for idempotent construction (trace-vanishing lemma).
• Wedderburn isomorphism for explicit basis construction of code ideals, using transversals of the group quotient and cyclic subgroup structure.
• Distance lower bounds derive from the block structure: any codeword not supported in at least two $K$-cosets must vanish within the central block.
• Non-equivalence proofs for non-central codes are established via explicit weight distributions and demonstration that no coordinate permutation can transform a refined non-central block into any abelian group code block.

The rigorous theory consolidates direct algebraic constructions with precise group-theoretic enumeration, confirming the richness and utility of metacyclic $p$-group codes for both combinatorial optimization and algebraic coding theory (Chahal et al., 5 Nov 2025, Vedenev, 16 Apr 2025, Yonglin et al., 2016).