Cyclic Convolutional Codes

• Cyclic Convolutional Codes are convolutional codes that incorporate cyclic symmetry via noncommutative skew polynomial rings.
• They are constructed through a mapping from vector polynomials to ideals in skew polynomial or Ore extension rings, enabling systematic encoder designs and decoding strategies.
• These codes achieve strong error-correction properties, linking optimal distance bounds with practical Reed–Solomon-based constructions and unit-memory families.

Searching arXiv for recent and foundational papers on cyclic convolutional codes to ground the article. Cyclic convolutional codes are convolutional codes endowed with a cyclic symmetry through a skew action of an algebra automorphism rather than through the ordinary cyclic shift. In the foundational formulation, with $A=\mathbb{F}_q[x]/(x^n-1)$ and $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$, a $\sigma$-cyclic convolutional code is a direct-summand $\mathbb{F}_q[z]$-submodule $C\subseteq \mathbb{F}_q[z]^n$ such that, under the identification $p:\mathbb{F}_q[z]^n\to A[z]$, the image $p(C)$ is a left ideal in the skew polynomial ring $R=A[z;\sigma]$ with multiplication rule $za=\sigma(a)z$ (0708.1343). A later extension replaces $A[z;\sigma]$ by Ore extensions $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$0 and skew Laurent series, yielding a general cyclic theory in the presence of a skew derivation and a right-ideal correspondence (Gómez-Torrecillas et al., 7 Jul 2025).

1. Definitions and conceptual framework

A convolutional code of length $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$1 and dimension $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$2 is an $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$3-submodule $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$4 of the form

$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$5

where $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$6 is basic, meaning $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$7 for all $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$8 in an algebraic closure of $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$9. Equivalently, $\sigma$0 is a direct summand of $\sigma$1. The degree of an encoder is

$\sigma$2

and for a minimal encoder this equals the overall constraint length $\sigma$3. A basic encoder is minimal if its row degrees sum to $\sigma$4; the row degrees of any minimal encoder are the Forney indices $\sigma$5, with $\sigma$6. For $\sigma$7, the weight is $\sigma$8, and the free distance is

$\sigma$9

(0708.1343).

The classical Laurent-series viewpoint defines a convolutional code of rate $\mathbb{F}_q[z]$0 over a field $\mathbb{F}_q[z]$1 as a $\mathbb{F}_q[z]$2-dimensional $\mathbb{F}_q[z]$3-vector subspace $\mathbb{F}_q[z]$4 that admits a basis by polynomial sequences; equivalently, $\mathbb{F}_q[z]$5 has an $\mathbb{F}_q[z]$6-basis. A key fact due to Roos is that $\mathbb{F}_q[z]$7 gives a bijection between codes $\mathbb{F}_q[z]$8 and $\mathbb{F}_q[z]$9-submodules of $C\subseteq \mathbb{F}_q[z]^n$0 that are direct summands, with rank equal to the code dimension (Gómez-Torrecillas et al., 7 Jul 2025).

The cyclic condition is nontrivial only after skewing the ambient algebra. For $C\subseteq \mathbb{F}_q[z]^n$1, classical cyclic block codes are ideals in the commutative ring $C\subseteq \mathbb{F}_q[z]^n$2 and correspond to convolutional codes of degree zero. If one demands invariance under the usual cyclic shift on $C\subseteq \mathbb{F}_q[z]^n$3, any such convolutional code must have degree zero. The skewed notion avoids this collapse: cyclicity is transferred from ordinary coordinate shift to closure under multiplication in a noncommutative skew polynomial ring (0708.1343).

2. Skew-polynomial realization of cyclicity

The standard cyclic setup fixes

$C\subseteq \mathbb{F}_q[z]^n$4

usually with $C\subseteq \mathbb{F}_q[z]^n$5, so that $C\subseteq \mathbb{F}_q[z]^n$6 splits into distinct linear factors and the Chinese remainder theorem yields

$C\subseteq \mathbb{F}_q[z]^n$7

($C\subseteq \mathbb{F}_q[z]^n$8 copies). Writing the primitive idempotents as $C\subseteq \mathbb{F}_q[z]^n$9, one often chooses an automorphism $p:\mathbb{F}_q[z]^n\to A[z]$0 acting as a single $p:\mathbb{F}_q[z]^n\to A[z]$1-cycle,

$p:\mathbb{F}_q[z]^n\to A[z]$2

The associated skew polynomial ring is

$p:\mathbb{F}_q[z]^n\to A[z]$3

which is noncommutative unless $p:\mathbb{F}_q[z]^n\to A[z]$4 (0708.1343).

The module identification between vector polynomials and skew polynomials is central. Define

$p:\mathbb{F}_q[z]^n\to A[z]$5

where

$p:\mathbb{F}_q[z]^n\to A[z]$6

If the commutative multiplication on $p:\mathbb{F}_q[z]^n\to A[z]$7 is replaced by the skew rule $p:\mathbb{F}_q[z]^n\to A[z]$8, then $p:\mathbb{F}_q[z]^n\to A[z]$9 identifies $p(C)$0 with $p(C)$1 as left $p(C)$2-modules. A submodule $p(C)$3 is $p(C)$4-cyclic precisely when $p(C)$5 is a left ideal in $p(C)$6 and $p(C)$7 is a direct summand. Such codes correspond to principal left ideals $p(C)$8 that are direct summands, and the components of $p(C)$9 govern the minimal encoder and the Forney indices (0708.1343).

A later generalization adopts right-module conventions. Let $R=A[z;\sigma]$0 be a finite-dimensional $R=A[z;\sigma]$1-algebra, $R=A[z;\sigma]$2, and $R=A[z;\sigma]$3 a $R=A[z;\sigma]$4-derivation satisfying

$R=A[z;\sigma]$5

Then the Ore extension $R=A[z;\sigma]$6 is defined by

$R=A[z;\sigma]$7

In this framework, cyclic convolutional codes are right $R=A[z;\sigma]$8-submodules of $R=A[z;\sigma]$9, and under appropriate nilpotency assumptions on $za=\sigma(a)z$0 and $za=\sigma(a)z$1, they correspond to right $za=\sigma(a)z$2-submodules of $za=\sigma(a)z$3 that are $za=\sigma(a)z$4-direct summands (Gómez-Torrecillas et al., 7 Jul 2025).

3. Matrix ring description and parameter extraction

When $za=\sigma(a)z$5 and $za=\sigma(a)z$6 acts as a full cycle on the primitive idempotents, the skew ring $za=\sigma(a)z$7 admits a concrete matrix realization over $za=\sigma(a)z$8, where $za=\sigma(a)z$9 corresponds to $A[z;\sigma]$0. The relevant ring is

$A[z;\sigma]$1

that is, the entries below the main diagonal are multiples of $A[z;\sigma]$2. The isomorphism $A[z;\sigma]$3 satisfies

$A[z;\sigma]$4

and

$A[z;\sigma]$5

Thus $A[z;\sigma]$6 is generated by a diagonal algebra and a single companion-type shift matrix with $A[z;\sigma]$7 in the lower-left position (0708.1343).

The Peirce decomposition

$A[z;\sigma]$8

provides the support

$A[z;\sigma]$9

Each component has the form

$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$00

A polynomial $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$01 is delay-free if $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$02 equals the support of its constant term; equivalently, if $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$03, then $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$04 for all $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$05. It is semi-reduced if the leading coefficients of its nonzero components lie in pairwise distinct ideals $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$06. Reducedness is stronger; semi-reducedness is sufficient for much of the coding theory developed in this setting (0708.1343).

The degree matrix makes the code invariants explicit. For $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$07 with $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$08, define

$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$09

(with $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$10 for zero entries). Then

$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$11

In each nontrivial row the entries are pairwise distinct, so there is a unique row maximum. The polynomial $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$12 is semi-reduced if and only if the maxima of all nontrivial rows occur in different columns (0708.1343).

If $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$13 is semi-reduced and delay-free with support $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$14, then the code $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$15 has dimension $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$16, and a minimal encoder is obtained directly from the supported components: $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$17 The Forney indices are

$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$18

and the overall constraint length is $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$19. Basicness is characterized both ring-theoretically and matrix-theoretically: a semi-reduced polynomial $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$20 is basic if and only if there exists a unit $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$21 such that $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$22; equivalently, the nonzero rows of $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$23 form a basic matrix over $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$24, meaning that its $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$25-minors are coprime (0708.1343).

4. Existence theory, constructive methods, and explicit families

The existence problem for prescribed Forney indices is reduced to a combinatorial matching condition. Define the $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$26 matrix

$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$27

Given desired Forney indices $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$28, write

$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$29

The modified rook problem asks whether there exist distinct row indices $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$30 and distinct column indices $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$31 such that

$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$32

If the rook problem is solvable for $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$33, then there exists a $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$34-dimensional $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$35-cyclic convolutional code with Forney indices $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$36. The construction proceeds by translating the placements $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$37 into target row degrees for a basic matrix over $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$38, extending to a matrix in $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$39, applying $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$40, and extracting the encoder from the supported components (0708.1343).

The constructive step is controlled by a degree-placement theorem: given pairwise distinct columns $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$41 and nonnegative degrees $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$42 with the constraint $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$43, there exists a basic matrix $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$44 whose row maxima appear in the prescribed columns and with prescribed degree inequalities across each row. Semi-reduction is achieved by left multiplication with finitely many elementary units of three types: $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$45 This guarantees that arbitrary matrices in $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$46 can be transformed to semi-reduced form without changing the left ideal (0708.1343).

The general solvability of the modified rook problem remains open, but it was verified computationally for $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$47. Two special cases are proved: solvability always holds if $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$48, and it also holds when the multiset $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$49 has at most two distinct values (0708.1343).

Several explicit families illustrate the theory.

Family	Parameters	Distance property
$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$50 with $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$51, $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$52	$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$53, Forney index $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$54	$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$55, generalized Singleton bound attained
$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$56	$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$57, $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$58, $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$59	$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$60, Griesmer bound attained for $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$61
$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$62 factor case	Example over $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$63, $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$64, $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$65, $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$66	Griesmer bound attained

For the one-dimensional family,

$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$67

and the code is MDS. For the unit-memory family with

$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$68

each block code generated by $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$69 and $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$70 is MDS with distance $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$71, implying $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$72. The construction generalizes to

$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$73

yielding $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$74-dimensional unit-memory codes with all Forney indices equal to $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$75 (0708.1343).

5. General cyclic convolutional codes with derivations and Laurent series

The Ore-extension framework broadens cyclic convolutional coding beyond the automorphism-only case. With $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$76 defined by $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$77, the main technical issue is the construction of a consistent skewed Laurent-series module structure on $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$78 when $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$79. The paper resolves this algebraically by first constructing skew formal power series $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$80 under local nilpotency of $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$81, and then localizing to skew Laurent series $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$82 when $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$83 is an automorphism and $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$84 is nilpotent (Gómez-Torrecillas et al., 7 Jul 2025).

Under these hypotheses there is a unique ring structure on Laurent series $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$85 satisfying the shift law

$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$86

and the inverse action

$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$87

where $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$88 is such that $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$89. The correspondence theorem states that if $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$90 is a finite-dimensional $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$91-algebra, $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$92, and $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$93 are nilpotent $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$94-derivations, then

$\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$95

is a bijection between $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$96-cyclic convolutional codes of rate $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$97 and right $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$98-submodules of $\sigma\in\operatorname{Aut}_{\mathbb{F}_q}(A)$99 that are $\sigma$00-direct summands of rank $\sigma$01. When $\sigma$02 as right $\sigma$03-modules, this becomes a bijection with right ideal codes in $\sigma$04 (Gómez-Torrecillas et al., 7 Jul 2025).

The theory also identifies obstructions. Without local nilpotency of $\sigma$05, the skew product on $\sigma$06 may fail to exist. Even with local nilpotency, $\sigma$07 may fail to be a right denominator set, so $\sigma$08 need not exist in a compatible Laurent extension. An explicit example is given by $\sigma$09 in characteristic $\sigma$10, $\sigma$11, and $\sigma$12 the usual derivation: here $\sigma$13 is a left denominator set but not right reversible, and there exists $\sigma$14 with $\sigma$15 (Gómez-Torrecillas et al., 7 Jul 2025).

This general setting does not assume that $\sigma$16 is a principal right ideal ring. Right ideals need not be principal, and the paper does not develop Forney indices or constraint lengths explicitly in the derivation case. Instead, it emphasizes the direct-summand condition and the right-ideal correspondence. Examples include $\sigma$17 with Frobenius automorphism and an inner $\sigma$18-derivation satisfying $\sigma$19, and a group-algebra example $\sigma$20 with $\sigma$21 cyclic of order $\sigma$22 and $\sigma$23 (Gómez-Torrecillas et al., 7 Jul 2025).

6. Special subclasses, decoding theory, related constructions, and open questions

A particularly structured subclass is formed by doubly cyclic convolutional codes. Here $\sigma$24, $\sigma$25, $\sigma$26 for a primitive $\sigma$27, and

$\sigma$28

For $\sigma$29, one defines matrices $\sigma$30 whose $\sigma$31-th row is $\sigma$32, and sets

$\sigma$33

The resulting code $\sigma$34 is cyclic because it corresponds to the left ideal generated by

$\sigma$35

in $\sigma$36, and it is “doubly cyclic” because the associated block codes $\sigma$37 are Reed–Solomon codes. The matrix $\sigma$38 is basic and reduced with all Forney indices equal to $\sigma$39, and

$\sigma$40

Moreover,

$\sigma$41

is a Reed–Solomon code of length $\sigma$42, dimension $\sigma$43, and minimum distance

$\sigma$44

(0908.0753).

The decoding theory of doubly cyclic codes exploits this embedded Reed–Solomon structure. With window size $\sigma$45 and step size $\sigma$46, the iterative algorithm decodes the first block of each window by subtracting the known state contribution, decoding a windowed block code $\sigma$47, and, in the doubly cyclic case, reducing the main step to Reed–Solomon decoding on the nested block codes $\sigma$48. If

$\sigma$49

then a formal correctness guarantee holds whenever every processed window contains at most $\sigma$50 symbol errors. The per-window complexity is

$\sigma$51

operations over $\sigma$52. For $\sigma$53, doubly cyclic codes are MDS and meet the generalized Singleton bound; for $\sigma$54, they attain the Griesmer bound (0908.0753).

Cyclic structure also appears in constructions with locality. A cyclic LRC code can be represented in quasicyclic form, and this QC representation can be mapped to a tail-biting convolutional code. If the QC generator blocks are converted into polynomials

$\sigma$55

the resulting convolutional generator $\sigma$56 has memory $\sigma$57, rate $\sigma$58, and constraint length $\sigma$59. Under the tail-biting symmetry $\sigma$60, the convolutional code equals the QC code modulo $\sigma$61, and the row codes inherit $\sigma$62 locality. The column-distance bound is

$\sigma$63

(Chen et al., 2020).

A different connection with cyclic block codes proceeds through Justesen’s encoder mapping. If $\sigma$64 is a cyclic code of length $\sigma$65, then under suitable root-distribution conditions the polynomial matrix

$\sigma$66

is a minimal basic encoder of a rate $\sigma$67 convolutional code with

$\sigma$68

This gives a systematic pathway from cyclic block codes of composite length to convolutional codes with large free distance (Xiong, 2017).

Several open problems remain central. In the matrix-ring theory, the general solvability of the modified rook problem for arbitrary residues $\sigma$69 remains open, despite computational evidence for $\sigma$70. Distance theory is also incomplete: beyond explicit families such as the one-dimensional MDS codes and some unit-memory constructions, no universal free-distance bounds specific to cyclic convolutional codes are derived. In the broader Ore-extension framework, principality of right ideals, existence of idempotent generators beyond the constant-coefficient case, and explicit computation of minimal encoders and Forney indices in the presence of a skew derivation remain open directions (0708.1343).