Skew Polycyclic Codes

• Skew polycyclic codes are linear codes defined as left ideals or submodules in noncommutative polynomial quotient rings, generalizing cyclic, constacyclic, and quasi-cyclic codes.
• They leverage the Ore skew polynomial ring framework with automorphisms and skew derivations to enable efficient algebraic decomposition and Gray map constructions.
• These codes facilitate advanced error-correcting techniques through designed distance, skew BCH/Reed–Solomon constructions, and principal ideal generators in various ring settings.

Searching arXiv for relevant papers on skew polycyclic codes and closely related skew cyclic/code-structure papers. Skew polycyclic codes are linear codes defined through noncommutative polynomial quotients, typically as left submodules or left ideals of rings of the form $S[t;\sigma,\delta]/\langle f(t)\rangle$, where $S$ is a ring, $\sigma$ is an automorphism, $\delta$ is a left $\sigma$-derivation, and $f(t)$ is monic of degree $m>1$ (Pumpluen, 13 Aug 2025). In the automorphism-only case one writes $F[x;\sigma]/(f)$, and the resulting objects are the noncommutative analogues of classical polycyclic codes over $F[x]/(f)$ (Gluesing-Luerssen, 2019). Skew cyclic codes, skew constacyclic codes, and skew quasi-cyclic variants are therefore special cases of a broader skew polycyclic framework; much of the literature develops this framework over finite fields, finite chain rings, semi-local non-chain rings, skew group rings, and even nonassociative ambient algebras (Ashraf et al., 2015, Islam et al., 2017, Pumpluen, 13 Aug 2025).

1. Algebraic framework

The foundational object is the Ore skew polynomial ring. For a unital ring $S$, an automorphism $S$0, and a left $S$1-derivation $S$2, the ring $S$3 consists of polynomials $S$4 with multiplication determined by

$S$5

When $S$6, this reduces to the automorphism-type skew polynomial ring $S$7; when $S$8 and $S$9, one recovers the usual commutative polynomial ring (Pumpluen, 13 Aug 2025). In the classical skew-cyclic literature this appears as $\sigma$0, with multiplication $\sigma$1, and a code of length $\sigma$2 is identified with a left submodule of $\sigma$3 for a monic polynomial $\sigma$4 of degree $\sigma$5 (Gluesing-Luerssen, 2019).

For

$\sigma$6

a right skew $\sigma$7-polycyclic code is a left linear code $\sigma$8 stable under the skew polycyclic shift

$\sigma$9

where $\delta$0 (Pumpluen, 13 Aug 2025). Special cases recover skew constacyclic codes when $\delta$1, skew cyclic codes when $\delta$2, and the commutative polycyclic setting when $\delta$3.

A common misconception is that skew polycyclic codes are essentially just skew cyclic codes. The general theory is broader: skew cyclic codes correspond to the special case $\delta$4, skew constacyclic codes to $\delta$5, and more general skew polycyclic codes arise from arbitrary central or otherwise admissible moduli $\delta$6 (Ashraf et al., 2015, Gluesing-Luerssen, 2019).

2. Ambient algebras and ideal-theoretic realization

A central structural idea is that skew polycyclic codes are realized as principal left ideals inside suitable quotient algebras. For a monic $\delta$7, the quotient

$\delta$8

can be endowed with a multiplication by taking right remainders modulo $\delta$9. This yields a Petit algebra, which is generally nonassociative unless $\sigma$0 is two-sided (Pumpluen, 13 Aug 2025). The code space $\sigma$1 is identified with $\sigma$2 via

$\sigma$3

and principal left ideals $\sigma$4 generated by monic right divisors $\sigma$5 of $\sigma$6 correspond to skew polycyclic codes of dimension $\sigma$7 (Pumpluen, 13 Aug 2025).

This nonassociative viewpoint is not merely formal. One explicit motivation is that allowing nonassociative ambient algebras eliminates the need on restrictions on the length of the codes that arise in associative quotients, especially for skew constacyclic codes over fields (Pumpluen, 13 Aug 2025). In the automorphism-only setting, the survey treatment of $\sigma$8-skew-cyclic codes gives the same principle in associative language: every left submodule of $\sigma$9 is generated by a unique monic right divisor of $f(t)$0 (Gluesing-Luerssen, 2019).

Alternative ambient realizations exist. Twisted skew $f(t)$1-codes are left ideals in twisted skew group rings $f(t)$2, and for cyclic $f(t)$3 these recover skew constacyclic codes; Proposition 2.3 shows that every twisted skew $f(t)$4-code is isomorphic to a skew $f(t)$5-constacyclic ring for some $f(t)$6 fixed by $f(t)$7 (Behajaina et al., 2022). Likewise, quotients of orders in cyclic division algebras satisfy

$f(t)$8

so skew polynomial codes arise naturally as left ideals in algebraic reduction rings connected with lattice constructions and space-time coding (Ducoat et al., 2015).

3. Decomposition methods and Gray maps

A recurrent theme in the ring-based literature is decomposition into field components. Over

$f(t)$9

every linear code $m>1$0 decomposes uniquely as

$m>1$1

with $m>1$2, and $m>1$3 is skew cyclic if and only if each $m>1$4 is skew cyclic over $m>1$5 with respect to the same automorphism $m>1$6 (Ashraf et al., 2015). The associated Gray map

$m>1$7

is an $m>1$8-linear isometry, and the Gray image of a skew cyclic code of length $m>1$9 is a skew $F[x;\sigma]/(f)$0-quasi-cyclic code of length $F[x;\sigma]/(f)$1 over $F[x;\sigma]/(f)$2 (Ashraf et al., 2015).

An analogous four-component picture holds over

$F[x;\sigma]/(f)$3

There, skew cyclic and skew $F[x;\sigma]/(f)$4-constacyclic codes decompose through orthogonal idempotents $F[x;\sigma]/(f)$5, and the Gray map sends skew cyclic codes to skew quasi-cyclic codes of length $F[x;\sigma]/(f)$6 over $F[x;\sigma]/(f)$7 of index $F[x;\sigma]/(f)$8, while skew $F[x;\sigma]/(f)$9-constacyclic codes map to skew quasi-twisted codes of index $F[x]/(f)$0 (Islam et al., 2017). These results are significant because they translate ring-linear skew polycyclic structure into field-linear QC/QT structure without losing distance information.

Over the rings $F[x]/(f)$1, Gray-like maps $F[x]/(f)$2 and $F[x]/(f)$3 similarly decompose $F[x]/(f)$4-cyclic codes into families of quasi-$F[x]/(f)$5-cyclic codes over $F[x]/(f)$6, with the number of components equal to $F[x]/(f)$7 (Irwansyah et al., 2017). Over $F[x]/(f)$8, $F[x]/(f)$9, the Gray map

$S$0

is an isometry from $S$1 to $S$2, and skew cyclic codes with derivation over $S$3 produce quaternary codes via Gray image, residue, torsion, and Plotkin sum constructions (Suprijanto et al., 2021).

These componentwise and Gray-map constructions suggest a general principle: skew polycyclic codes over structured rings are often most tractable after passage to field components, where cyclic, quasi-cyclic, or quasi-twisted phenomena become explicit (Ashraf et al., 2015, Irwansyah et al., 2017, Islam et al., 2017).

4. Generators, idempotents, duality, and refined structure theorems

Principal generation is one of the strongest recurring structural facts. Over $S$4, every skew cyclic code is principally generated by

$S$5

where the $S$6 are the monic generator polynomials of the component skew cyclic codes over $S$7, and $S$8 is a right divisor of $S$9 in $S$00 (Ashraf et al., 2015). Over $S$01, the same phenomenon holds both for skew cyclic codes and skew $S$02-constacyclic codes: every left submodule of $S$03 and every left submodule of $S$04 is principal (Islam et al., 2017).

Idempotent generators appear under arithmetic hypotheses. For skew cyclic codes over $S$05, if $S$06 and $S$07, then each component code has an idempotent generator, and these combine into an idempotent generator $S$08 of the full code in $S$09 (Ashraf et al., 2015). Parallel results hold for skew cyclic and skew constacyclic codes over $S$10 (Islam et al., 2017). In twisted skew group rings, idempotents also govern Euclidean LCD and self-dual structures: a twisted skew group code $S$11 is Euclidean LCD iff $S$12, and Euclidean self-dual iff $S$13 together with $S$14 (Behajaina et al., 2022).

Duality is explicit in several settings. For skew constacyclic codes in $S$15, the Euclidean dual of a $S$16-constacyclic code is again $S$17-constacyclic, and its generator polynomial is computed from the parity-check polynomial via a skew-reciprocal construction (Ducoat et al., 2015). For automorphism-type skew constacyclic codes over fields, the survey literature gives parity-check and dual generator descriptions through skew circulants and left reciprocals (Gluesing-Luerssen, 2019). Over ring extensions such as $S$18 and $S$19, duality reduces componentwise to the field case (Ashraf et al., 2015, Islam et al., 2017).

Recent work over the chain ring

$S$20

pushes this structural analysis further. For central $S$21, the left ideals of $S$22 are classified in three general types: trivial ideals, ideals contained in $S$23, and ideals not contained in $S$24, each expressed through successive $S$25-layers and right divisors in the residue skew polynomial quotient (Tiwari et al., 13 May 2026). In the repeated-root skew constacyclic cases $S$26 with $S$27, the case $S$28 yields a simpler form of generators and a more refined structural characterization (Tiwari et al., 13 May 2026).

5. Designed distance, skew BCH and skew Reed–Solomon constructions

The distance theory of skew polycyclic codes is built from right roots and skew Vandermonde structures. In the survey treatment of $S$29, a monic $S$30 of degree $S$31 defines a $S$32-skew-cyclic code, and W-polynomials, skew Vandermonde matrices, and root sets yield parity-check representations and designed-distance constructions (Gluesing-Luerssen, 2019). The resulting skew BCH codes and skew Reed–Solomon codes are best understood as highly structured skew polycyclic codes.

For skew Reed–Solomon codes over $S$33, with $S$34 and $S$35 for a normal basis generator $S$36, the generator polynomial is

$S$37

and the code has length $S$38, dimension $S$39, and Hamming distance exactly $S$40; hence it is MDS (Gómez-Torrecillas et al., 2017). The noncommutative Peterson–Gorenstein–Zierler algorithm developed for these skew RS codes computes syndromes, builds an error-locator polynomial as a fully $S$41-decomposable polynomial, and recovers error positions and values both for block codes and for convolutional codes (Gómez-Torrecillas et al., 2017).

A broad Hartmann–Tzeng bound is available for skew cyclic codes. If the $S$42-defining set $S$43 of the generator contains

$S$44

with $S$45 and $S$46, then the minimum Hamming distance satisfies

$S$47

(Gómez-Torrecillas et al., 2017). The BCH bound appears as the special case $S$48. The same paper gives a practical construction method: one forms the prescribed index set, closes it under the relevant cosets in $S$49, and constructs the generator polynomial as the least common left multiple of the corresponding linear factors in an extension ring before contracting back to the base ring (Gómez-Torrecillas et al., 2017).

Applications of root-pattern arguments extend beyond distance bounds. Reversible DNA codes obtained from skew cyclic codes are characterized by palindromic or $S$50-palindromic generator polynomials, depending on the parity of the degree, and the dual of a reversible DNA code is again reversible (Gursoy et al., 2017). This use of generator symmetry is specific to the DNA setting, but it illustrates how right-divisor structure and skew reciprocal operations can encode combinatorial constraints beyond Hamming distance.

6. Classification, equivalence, and broader directions

A major recent development is the classification of skew polycyclic codes up to equivalence and isometry through ambient algebra isomorphisms. For classes $S$51 of skew $S$52-polycyclic codes, algebra isomorphisms

$S$53

preserve Hamming weight and map generators of principal left ideals to generators of principal left ideals (Pumpluen, 13 Aug 2025). This yields refined notions of isometry and equivalence that reduce the number of previously known isometry and equivalence classes, state precisely when different notions coincide, and classify families with the same performance parameters so as to avoid duplicating already existing codes (Pumpluen, 13 Aug 2025).

The paper also shows that nonassociative ambient algebras are not a pathology but a systematic feature: by working in Petit algebras rather than only associative quotients, one eliminates the need on restrictions on the length of the codes (Pumpluen, 13 Aug 2025). In many field cases with sufficiently large $S$54, isometry and equivalence coincide, and the classification problem reduces to explicit coefficient conditions involving norm-like expressions $S$55 (Pumpluen, 13 Aug 2025).

This classification agenda connects to several adjacent developments. Twisted skew $S$56-codes unify group codes, twisted group codes, and skew group codes as left ideals in $S$57, with cyclic $S$58 recovering skew constacyclic codes (Behajaina et al., 2022). Codes from quotients of cyclic division algebras connect skew polynomial ideals to lattices through a variation of Construction A (Ducoat et al., 2015). Over $S$59, Euclidean self-dual $S$60-cyclic codes correspond under Gray maps to families of Euclidean self-dual quasi-$S$61-cyclic codes over $S$62 (Irwansyah et al., 2017). Over $S$63, skew cyclic codes with derivation generate new linear codes over $S$64 with good parameters through Gray images, residue and torsion codes, and Plotkin sums (Suprijanto et al., 2021).

A further point of correction in the recent literature concerns the necessity of extra divisibility constraints in small-$S$65 repeated-root classifications. For $S$66, the 2026 analysis shows that some earlier descriptions omitted conditions needed to keep different classes of left ideals mutually disjoint; in the cases $S$67 and $S$68, these missing conditions are made explicit and certain $S$69-th torsion codes are computed (Tiwari et al., 13 May 2026). This suggests that classification over finite chain rings is now moving from broad existence theorems toward fully disjoint, computation-ready normal forms.

Taken together, these results show that skew polycyclic codes are not a single isolated family but a unifying language for cyclic-type coding over noncommutative polynomial quotients. Their theory combines right-divisor factorization, left-ideal structure, Gray-map transfer, idempotent decomposition, designed-distance constructions, and ambient-algebra isometries in a way that simultaneously generalizes cyclic, constacyclic, quasi-cyclic, and several group-theoretic code classes (Gluesing-Luerssen, 2019, Pumpluen, 13 Aug 2025).