Monotone Chain Polar Codes

• Monotone chain polar codes are defined by a total ordering of polarized symbols that is monotone within each component, enabling full rate region coverage in multiterminal Slepian–Wolf problems.
• They are algebraically modeled as decreasing monomial-Cartesian codes, offering explicit formulas for parameters like length, dimension, and minimum distance.
• The decoding strategies, including successive cancellation and list decoding, are optimized according to chain structure, balancing complexity and performance improvements over classical polar codes.

Monotone chain polar codes are a class of polar-code constructions in which the polarized symbols are decoded according to a total ordering that is monotone within each component sequence, and they appear in two closely related research lines: as multivariate polar constructions for distributed lossless coding, and as algebraic instances of polar decreasing monomial-Cartesian codes over finite fields (Ren et al., 3 Sep 2025, Camps et al., 2020). In the multivariate viewpoint, a monotone chain is a decoding order on the joint collection of polarized variables from multiple terminals, chosen so that each terminal’s own copy-indices remain in natural order; by varying this chain, one can realize the entire admissible rate region on the dominant face of the Slepian–Wolf problem without time-sharing (Ren et al., 3 Sep 2025). In the algebraic viewpoint, polar-code information sets defined by thresholding monomials under a preorder refining divisibility yield decreasing monomial-Cartesian codes, thereby placing classical binary polar codes, multikernel constructions, and finite-field Reed–Solomon-kernel variants in a common framework (Camps et al., 2020).

1. Algebraic formulation via decreasing monomial-Cartesian codes

Let $\mathbb{F}=\mathbb{F}_q$ be a finite field, and let

$R=\mathbb{F}[x_1,\dots,x_m]$

be the polynomial ring in $m$ variables. Fix nonempty subsets $S_i\subseteq\mathbb{F}$ of sizes $n_i:=|S_i|$, and form the Cartesian set

$$S=S_1\times\cdots\times S_m=\{s^{(1)},\dots,s^{(n)}\}\subseteq \mathbb{F}^m,\qquad n=\prod_{i=1}^m n_i,$$

with an arbitrary but fixed linear ordering on $S$. A set $\Delta$ of monomials in $R$ is called closed under divisibility, or decreasing, if whenever $M\in\Delta$ and $R=\mathbb{F}[x_1,\dots,x_m]$0, then $R=\mathbb{F}[x_1,\dots,x_m]$1 (Camps et al., 2020).

Writing $R=\mathbb{F}[x_1,\dots,x_m]$2 for the $R=\mathbb{F}[x_1,\dots,x_m]$3-span of $R=\mathbb{F}[x_1,\dots,x_m]$4, the evaluation map

$R=\mathbb{F}[x_1,\dots,x_m]$5

is $R=\mathbb{F}[x_1,\dots,x_m]$6-linear. Under the assumption that every monomial in $R=\mathbb{F}[x_1,\dots,x_m]$7 has $R=\mathbb{F}[x_1,\dots,x_m]$8-degree $R=\mathbb{F}[x_1,\dots,x_m]$9, the map is injective. The associated decreasing monomial-Cartesian code is

$m$0

(Camps et al., 2020).

This formulation is the algebraic backbone of the monotone-chain connection. The key point is that a polar-code information set can be indexed by monomials, and when that set is selected by a threshold in an order refining divisibility, the resulting code is a decreasing monomial-Cartesian code. A plausible implication is that many design questions for monotone-chain polar codes can be rephrased as structural questions about monomial ideals on Cartesian point sets.

2. Partial orders, threshold monomials, and the monotone-chain viewpoint

To isolate the polar information sets among decreasing monomial sets, Camps, López, Matthews, and Sarmiento introduce a partial order on monomials inspired by Bardet, Dragoi, Otmani, and Tillich. For monomials

$m$1

one declares $m$2 if one of three conditions holds: divisibility $m$3; a tie-break in one variable block when $m$4 but $m$5, comparing exponent vectors $m$6 and $m$7 in the product order; or a recursive extension across two variable blocks, where the first-block projections satisfy the same order and the second block is arbitrary (Camps et al., 2020).

In practice, this preorder refines divisibility and is compatible with the usual Kronecker-product construction of polar kernels. Whenever the information set of the polar-code construction from a kernel sequence $m$8 is taken to be those row-monomials $m$9 satisfying $S_i\subseteq\mathbb{F}$0 for some threshold monomial $S_i\subseteq\mathbb{F}$1, the information set forms a divisibility-closed chain and recovers exactly a decreasing monomial-Cartesian code (Camps et al., 2020).

This is the point at which monotone-chain polar codes enter the algebraic picture. In the distributed-coding literature, a monotone chain is a total ordering of polarized variables from multiple sources; in the monomial-Cartesian literature, the corresponding information set is specified by a preorder threshold on monomials. The shared feature is a structured monotonicity constraint: order is unrestricted globally, but restricted locally so that divisibility or per-source index order is preserved.

3. Polarization over finite fields and multikernel constructions

For a discrete memoryless channel $S_i\subseteq\mathbb{F}$2, the finite-field framework distinguishes additive-symmetric and multiplicative-symmetric channels. The channel is additive-symmetric if for every $S_i\subseteq\mathbb{F}$3 there is a permutation $S_i\subseteq\mathbb{F}$4 of $S_i\subseteq\mathbb{F}$5 such that

$S_i\subseteq\mathbb{F}$6

and multiplicative-symmetric if for every $S_i\subseteq\mathbb{F}$7 there is a permutation $S_i\subseteq\mathbb{F}$8 of $S_i\subseteq\mathbb{F}$9 such that

$n_i:=|S_i|$0

If both conditions hold, the channel is symmetric over the field, or SOF (Camps et al., 2020).

Given invertible matrices $n_i:=|S_i|$1 of sizes $n_i:=|S_i|$2, the multikernel construction uses

$n_i:=|S_i|$3

up to bit-reversal. The sequence $n_i:=|S_i|$4 polarizes $n_i:=|S_i|$5 if, for all $n_i:=|S_i|$6, the split channels $n_i:=|S_i|$7 satisfy

$n_i:=|S_i|$8

The stated sufficient condition is that, if $n_i:=|S_i|$9 is the characteristic of $$S=S_1\times\cdots\times S_m=\{s^{(1)},\dots,s^{(n)}\}\subseteq \mathbb{F}^m,\qquad n=\prod_{i=1}^m n_i,$$0 and each $$S=S_1\times\cdots\times S_m=\{s^{(1)},\dots,s^{(n)}\}\subseteq \mathbb{F}^m,\qquad n=\prod_{i=1}^m n_i,$$1 has a non-identity standard form $$S=S_1\times\cdots\times S_m=\{s^{(1)},\dots,s^{(n)}\}\subseteq \mathbb{F}^m,\qquad n=\prod_{i=1}^m n_i,$$2 whose entries generate the full field,

$$S=S_1\times\cdots\times S_m=\{s^{(1)},\dots,s^{(n)}\}\subseteq \mathbb{F}^m,\qquad n=\prod_{i=1}^m n_i,$$3

then the sequence polarizes every SOF channel over $$S=S_1\times\cdots\times S_m=\{s^{(1)},\dots,s^{(n)}\}\subseteq \mathbb{F}^m,\qquad n=\prod_{i=1}^m n_i,$$4 (Camps et al., 2020).

A particularly important instance is the generalized Reed–Solomon evaluation matrix on any $$S=S_1\times\cdots\times S_m=\{s^{(1)},\dots,s^{(n)}\}\subseteq \mathbb{F}^m,\qquad n=\prod_{i=1}^m n_i,$$5 points of $$S=S_1\times\cdots\times S_m=\{s^{(1)},\dots,s^{(n)}\}\subseteq \mathbb{F}^m,\qquad n=\prod_{i=1}^m n_i,$$6,

$$S=S_1\times\cdots\times S_m=\{s^{(1)},\dots,s^{(n)}\}\subseteq \mathbb{F}^m,\qquad n=\prod_{i=1}^m n_i,$$7

The original Arıkan $$S=S_1\times\cdots\times S_m=\{s^{(1)},\dots,s^{(n)}\}\subseteq \mathbb{F}^m,\qquad n=\prod_{i=1}^m n_i,$$8 kernel $$S=S_1\times\cdots\times S_m=\{s^{(1)},\dots,s^{(n)}\}\subseteq \mathbb{F}^m,\qquad n=\prod_{i=1}^m n_i,$$9 yields the binary decreasing-monomial codes of Bardet et al., while Reed–Solomon kernels appear by taking the $S$0 to be $S$1 distinct points in $S$2; the summary states that they achieve the best possible exponent

$S$3

known in the literature and polarize SOF channels under the finite-field theorem (Camps et al., 2020).

The algebraic unification is therefore not limited to a single kernel family. It subsumes classical binary polar coding and extends it to arbitrary field sizes and arbitrary sequences of kernels satisfying the field-generation condition.

4. Code parameters, minimal generators, and duality

A subset $S$4 is a generating set if every monomial $S$5 divides some $S$6, and it is minimal if no two elements of $S$7 are divisible by one another. Every decreasing $S$8 has a unique minimal generator $S$9 (Camps et al., 2020). This object controls the basic parameters of the code.

The length is

$\Delta$0

If each $\Delta$1 is written as $\Delta$2, then the dimension is given by the inclusion–exclusion formula

$\Delta$3

where for each nonempty $\Delta$4,

$\Delta$5

The minimum distance is

$\Delta$6

(Camps et al., 2020).

The same work proves a duality theorem: the Euclidean dual $\Delta$7 is monomially equivalent to another decreasing monomial-Cartesian code $\Delta$8, where

$\Delta$9

Moreover, a basis of the dual can be written in terms of residue-polynomials, or Lagrange residues, attached to the monomials in $R$0 (Camps et al., 2020).

For monotone-chain polar codes, these formulas are significant because they make length, dimension, and minimum distance explicit in terms of the thresholded monomial set. The summary further states that the minimal-generator formula for the minimum distance shows that the minimum distance of the $R$1-fold Kronecker product code grows super-polynomially in $R$2 as soon as one picks information monomials all of high total degree, yielding explicit lower bounds on $R$3 versus $R$4 that in many cases exceed those known for random binary kernels of the same rate (Camps et al., 2020). It also states that the dual-code equivalence yields explicit LCD polar codes by exchanging $R$5.

5. Multiterminal monotone chains and the Slepian–Wolf rate region

In the distributed lossless coding formulation, let $R$6 be the number of correlated sources, with alphabet sizes $R$7, and let $R$8. For $R$9, define

$M\in\Delta$0

with each sequence an i.i.d. copy of source $M\in\Delta$1. The polar transform

$M\in\Delta$2

is applied separately to each source:

$M\in\Delta$3

The joint system contains the $M\in\Delta$4 polarized symbols $M\in\Delta$5, where $M\in\Delta$6 is a permutation of $M\in\Delta$7 (Ren et al., 3 Sep 2025).

A chain is a total ordering $M\in\Delta$8, and it is monotone if, whenever $M\in\Delta$9 with $R=\mathbb{F}[x_1,\dots,x_m]$00, one has $R=\mathbb{F}[x_1,\dots,x_m]$01. Equivalently, each source’s own copy-indices appear in natural order, so the monotone chain is fully specified by the length-$R=\mathbb{F}[x_1,\dots,x_m]$02 sequence $R=\mathbb{F}[x_1,\dots,x_m]$03 (Ren et al., 3 Sep 2025).

The entropy-chain-rule formulation is

$R=\mathbb{F}[x_1,\dots,x_m]$04

For each position define

$R=\mathbb{F}[x_1,\dots,x_m]$05

As in classical polar coding, these conditional entropies polarize to $R=\mathbb{F}[x_1,\dots,x_m]$06 or full $R=\mathbb{F}[x_1,\dots,x_m]$07. Fixing $R=\mathbb{F}[x_1,\dots,x_m]$08 and taking $R=\mathbb{F}[x_1,\dots,x_m]$09 large, one chooses a high-entropy set $R=\mathbb{F}[x_1,\dots,x_m]$10 of size approximately $R=\mathbb{F}[x_1,\dots,x_m]$11, where $R=\mathbb{F}[x_1,\dots,x_m]$12 corresponds to large $R=\mathbb{F}[x_1,\dots,x_m]$13 and thus frozen symbols, while $R=\mathbb{F}[x_1,\dots,x_m]$14 corresponds to small $R=\mathbb{F}[x_1,\dots,x_m]$15 and thus information symbols (Ren et al., 3 Sep 2025).

The achievable-rate statement is that, in the Slepian–Wolf problem, one seeks $R=\mathbb{F}[x_1,\dots,x_m]$16 on the dominant face of

$R=\mathbb{F}[x_1,\dots,x_m]$17

and Bilkent–Arıkan showed that by choosing different monotone chains $R=\mathbb{F}[x_1,\dots,x_m]$18, the induced rates

$R=\mathbb{F}[x_1,\dots,x_m]$19

cover every point on the dominant face, without time-sharing (Ren et al., 3 Sep 2025). The same chain rule and monotonicity argument extend to arbitrary $R=\mathbb{F}[x_1,\dots,x_m]$20 and $R=\mathbb{F}[x_1,\dots,x_m]$21, with quasigroups or random permutations used to ensure polarization in the non-binary setting.

6. Successive cancellation decoding, complexity, and list decoding

General monotone-chain polar codes introduce decoding issues that do not arise in the classical one-dimensional chain. The successive cancellation task is to estimate $R=\mathbb{F}[x_1,\dots,x_m]$22 sequentially for $R=\mathbb{F}[x_1,\dots,x_m]$23, given the previously decoded symbols, by computing

$R=\mathbb{F}[x_1,\dots,x_m]$24

Once $R=\mathbb{F}[x_1,\dots,x_m]$25 is chosen, whether frozen or by an ML rule, a partial decision is imposed at the corresponding leaf by making the probability tensor deterministic in the known components and uniform over the unknown ones (Ren et al., 3 Sep 2025).

The decoding framework in (Ren et al., 3 Sep 2025) represents any multivariate distribution $R=\mathbb{F}[x_1,\dots,x_m]$26 by an $R=\mathbb{F}[x_1,\dots,x_m]$27-dimensional tensor $R=\mathbb{F}[x_1,\dots,x_m]$28 and uses three tensor operations: circular convolution $R=\mathbb{F}[x_1,\dots,x_m]$29, dual convolution $R=\mathbb{F}[x_1,\dots,x_m]$30, and normalized product $R=\mathbb{F}[x_1,\dots,x_m]$31. On a size-$R=\mathbb{F}[x_1,\dots,x_m]$32 butterfly with relations $R=\mathbb{F}[x_1,\dots,x_m]$33 and $R=\mathbb{F}[x_1,\dots,x_m]$34, the inference updates are

$R=\mathbb{F}[x_1,\dots,x_m]$35

$R=\mathbb{F}[x_1,\dots,x_m]$36

$R=\mathbb{F}[x_1,\dots,x_m]$37

The full polar transform is viewed as a directed computation graph with $R=\mathbb{F}[x_1,\dots,x_m]$38 edges and $R=\mathbb{F}[x_1,\dots,x_m]$39 vertices, each edge carrying a probability tensor and each vertex implementing one of these local update rules (Ren et al., 3 Sep 2025).

Two complexity facts are central. First, any SC decoder must spend $R=\mathbb{F}[x_1,\dots,x_m]$40 operations, since each of the $R=\mathbb{F}[x_1,\dots,x_m]$41 leaves must receive the root prior at least once along a path of length $R=\mathbb{F}[x_1,\dots,x_m]$42. Second, a general monotone chain requires at most $R=\mathbb{F}[x_1,\dots,x_m]$43 work, because each of the $R=\mathbb{F}[x_1,\dots,x_m]$44 decoding steps may force a recomputation along the path between successive leaves, which costs $R=\mathbb{F}[x_1,\dots,x_m]$45 in the worst case (Ren et al., 3 Sep 2025). Corner-point chains, in which one decodes each source in a single block, reduce to classical SC in each block and therefore achieve total $R=\mathbb{F}[x_1,\dots,x_m]$46 complexity; alternating chains can force leaf jumps across the root at every step and therefore incur $R=\mathbb{F}[x_1,\dots,x_m]$47 complexity.

A common misconception is that the classical $R=\mathbb{F}[x_1,\dots,x_m]$48-space optimization for SC decoding extends unchanged to general monotone chains. The 2025 decoding paper states explicitly that this is not universally applicable: in a general monotone chain, frequent back-and-forth between leaves invalidates one-time deletion of graph memory, so one must keep $R=\mathbb{F}[x_1,\dots,x_m]$49 tensors or accept $R=\mathbb{F}[x_1,\dots,x_m]$50 time through recomputation (Ren et al., 3 Sep 2025).

For list decoding, the same work introduces a pointer-based computational-graph implementation with two core data structures, Edge and Vertex, and a head pointer decHead. The initGraph() procedure allocates all $R=\mathbb{F}[x_1,\dots,x_m]$51 edges, decodeAt(t) moves the head along a path of vertices in $R=\mathbb{F}[x_1,\dots,x_m]$52, and constant-time decoder forking is achieved by copying only the decHead pointer. This enables time-efficient list decoding without relying on $R=\mathbb{F}[x_1,\dots,x_m]$53-space tricks (Ren et al., 3 Sep 2025).

The reported runtime comparison for list size $R=\mathbb{F}[x_1,\dots,x_m]$54, corner chain, and binary sources shows the following timings:

Blocklength	Lazy-copy time (s)	Proposed (s)
$R=\mathbb{F}[x_1,\dots,x_m]$55	$R=\mathbb{F}[x_1,\dots,x_m]$56	$R=\mathbb{F}[x_1,\dots,x_m]$57
$R=\mathbb{F}[x_1,\dots,x_m]$58	$R=\mathbb{F}[x_1,\dots,x_m]$59	$R=\mathbb{F}[x_1,\dots,x_m]$60
$R=\mathbb{F}[x_1,\dots,x_m]$61	$R=\mathbb{F}[x_1,\dots,x_m]$62	$R=\mathbb{F}[x_1,\dots,x_m]$63
$R=\mathbb{F}[x_1,\dots,x_m]$64	$R=\mathbb{F}[x_1,\dots,x_m]$65	$R=\mathbb{F}[x_1,\dots,x_m]$66
$R=\mathbb{F}[x_1,\dots,x_m]$67	$R=\mathbb{F}[x_1,\dots,x_m]$68	$R=\mathbb{F}[x_1,\dots,x_m]$69
$R=\mathbb{F}[x_1,\dots,x_m]$70	$R=\mathbb{F}[x_1,\dots,x_m]$71	$R=\mathbb{F}[x_1,\dots,x_m]$72
$R=\mathbb{F}[x_1,\dots,x_m]$73	$R=\mathbb{F}[x_1,\dots,x_m]$74	$R=\mathbb{F}[x_1,\dots,x_m]$75

The corresponding improvements reported in the source range from $R=\mathbb{F}[x_1,\dots,x_m]$76 at $R=\mathbb{F}[x_1,\dots,x_m]$77 to $R=\mathbb{F}[x_1,\dots,x_m]$78 at $R=\mathbb{F}[x_1,\dots,x_m]$79, averaged over $R=\mathbb{F}[x_1,\dots,x_m]$80 runs with no fast-pruning or min-sum optimizations (Ren et al., 3 Sep 2025). For a non-binary example with $R=\mathbb{F}[x_1,\dots,x_m]$81, ternary $R=\mathbb{F}[x_1,\dots,x_m]$82, quinary $R=\mathbb{F}[x_1,\dots,x_m]$83, and a joint pmf of size $R=\mathbb{F}[x_1,\dots,x_m]$84, the same source reports that BLER improves dramatically when moving from SC ($R=\mathbb{F}[x_1,\dots,x_m]$85) to list decoding ($R=\mathbb{F}[x_1,\dots,x_m]$86, and that some non-corner chains slightly outperform corner chains at the same sum-rate.

Taken together, these results present monotone-chain polar codes as both an algebraic and an algorithmic generalization of classical polar coding: algebraically, they are captured by decreasing monomial-Cartesian codes and their duals; operationally, they achieve the full dominant face of the multiterminal Slepian–Wolf region and require decoding methods whose complexity depends sensitively on chain structure (Camps et al., 2020, Ren et al., 3 Sep 2025).