PAL-Set: Algebraic Moulds & Polar Codes

• PAL-Set is a concept defined in mould calculus and polar coding, capturing algebraic symmetries and sharply polarized channel behaviors.
• It establishes a bridge between Ecalle's double shuffle framework and polar code index sets through rigorous automorphism mappings.
• The analytic alignment and closure properties of PAL-Set enhance both theoretical insights into multiple zeta values and practical channel coding design.

The term "PAL-Set" appears in multiple advanced mathematical and information-theoretic contexts. The most prominent formalization is in the algebraic theory of moulds associated with Ecalle's framework for double shuffle relations of multiple zeta values (MZVs), as well as in polar coding theory, where it refers to sets of indices with sharply polarized behavior under the polar transform.

1. Algebraic Definition in Ecalle's Mould Calculus

The PAL-Set in this context is associated with the "pal" moulds within Ecalle’s algebraic formalism. Let $T$ be a finite abelian group and $\mathrm{BIMU}(T)$ the algebra of moulds with values in rational functions over $T$, equipped with the non-commutative product "$\times$" (Komiyama, 2021). Moulds are endowed with symmetry properties:

• Alternal: For all words $u, v$,

$\sum_{w \in \operatorname{Sh}(u, v)} M(w) = 0$

• Symmetral: For the same $u, v$,

$$\sum_{w \in \operatorname{Sh}(u, v)} M(w) = M(u) M(v)$$

• Analogous definitions obtain for "alternil" and "symmetrile," via stuffle (quasi-shuffle) operators.

The PAL-Set is the one-parameter family of "pal" moulds:

$$\operatorname{pal}(u_1,\ldots,u_m) = u_1 (u_1 + u_2)\cdots(u_1 + \cdots + u_m)$$

for $m\geq 1$ (with $\operatorname{pal}(\emptyset)=1$), satisfying the symmetrality property:

$$\sum_{w\in\operatorname{Sh}(u, v)} \operatorname{pal}(w)=\operatorname{pal}(u)\operatorname{pal}(v)$$

for words $u, v$ of arbitrary lengths.

2. PAL-Set as Structural Object and Automorphism Bridge

The "pal" mould is canonically attached to the exponential:

$$\operatorname{pal} = \exp^\times(A),\qquad A \in \mathrm{ARI}(T)_{\text{al}}$$

where $\mathrm{ARI}(T)_{\text{al}}$ denotes the Lie algebra of alternal moulds and $\exp^\times$ denotes the group exponential in $(\mathrm{GARI}(T),\times)$. The adjoint map

$$\operatorname{adari}(\operatorname{pal}) = \exp(\operatorname{arit}(\operatorname{pal}))$$

acts as a Lie algebra automorphism $$\operatorname{adari}(\operatorname{pal})\colon\mathrm{ARI}(T)_{\text{al}} \to \mathrm{ARI}(T)_{\text{il}}$$ and as a group automorphism on symmetral and symmetrile moulds (Komiyama, 2021). Thus, the PAL-Set is the image of the alternal Lie algebra under $\operatorname{adari}(\operatorname{pal})$, i.e., $\mathrm{ARI}(T)_{\text{il}}$, and similarly for groups.

This automorphism system establishes a bijection between symmetral/alternil and symmetrile/alternal classes, mapping the “all-shuffles” side of the double shuffle algebra onto the “all-stuffles” side.

3. Dimensional, Closure, and Generation Properties

By Theorem 3.7 of (Komiyama, 2021),

• $\operatorname{adari}(\operatorname{pal})$ defines an automorphism of both $\mathrm{ARI}(T)$ and $\mathrm{GARI}(T)$, preserving Lie and group structure.
• The subspace $$\mathrm{ARI}(T)_{\text{il}} = \operatorname{adari}(\operatorname{pal})(\mathrm{ARI}(T)_{\text{al}})$$ is itself a Lie subalgebra.
• Every element in $\mathrm{ARI}(T)_{\text{il}}$ can be generated under the $\text{ari}$-bracket by images of primitive polynomials.
• For any depth $r$, $$\dim \mathrm{ARI}(T)_{\text{il}} = \dim \mathrm{ARI}(T)_{\text{al}}$$, matching the number of Lyndon words with two letters, as conjectured for depth-$r$ components of the double-shuffle Lie algebra of MZVs.

4. PAL-Set in Polar Coding: Polarized Index Sets

In the context of polar codes, "PAL-Set" refers informally to the set of indices or synthesized channels exhibiting strong polarization—either essentially noiseless or almost perfectly noisy—under Arıkan’s polar transform. For a binary-input DMC $W$, the polar transform synthesizes $N=2^n$ bit-channels $W_N^{(i)}$, and via the Bhattacharyya parameter $Z(W)$, two principal sets are defined:

• Reliable indices: $$\mathcal{I}_\varepsilon(W) = \{i : Z(W_N^{(i)}) \leq \varepsilon \}$$
• Noisy indices: $$\mathcal{F}_\varepsilon(W) = \{i : Z(W_N^{(i)}) \geq 1-\varepsilon \}$$

Alignment or inclusion relations among the "reliable" sets for various channels are crucial for universality analysis and network coding applications (Renes et al., 2014).

5. Analytic Alignment Conditions and Applications

Two analytic conditions govern the alignment of PAL-Sets in polar code theory:

• Theorem 1 (Nonalignment): If for some path $b$,

$1 - I(W_b) + I(V_b) \geq 1$

then $$|\mathcal{F}_\varepsilon(W) \cap \mathcal{I}_\varepsilon(V)| = \Omega(N)$$, indicating non-alignment.

• Theorem 2 (Alignment): If

$Z(W_b) + Z((V^c)_{\bar b}) \leq 1$

holds for all relevant $b$, then $$\mathcal{F}_\varepsilon(W) \overset{\cdot}{\subseteq} \mathcal{F}_\varepsilon(V)$$ and $$\mathcal{I}_\varepsilon(V) \overset{\cdot}{\subseteq} \mathcal{I}_\varepsilon(W)$$.

These conditions facilitate precise analysis of universality of coding, secret-key requirements in wiretap channel models, and the necessity of entanglement assistance in quantum polar codes (Renes et al., 2014).

6. Connections to Double Shuffle and Universality

In mould theory, the PAL-Set provides a rigorous algebraic realization of the (alternil/symmetrile) side of the double shuffle symmetry expected in MZVs. In coding theory, PAL-Set alignment ensures the transfer of universality or converse results between classes of channels, supporting efficient code constructions for network, wiretap, and quantum scenarios.

A commutative diagram formalizes the relationship between PAL-Set and exponentials:

$\begin{array}{ccc} GARI(T)_{as} &\xrightarrow{\;\adari(\mathrm{pal})\;} &GARI(T)_{is} \ \exp^{\times}\downarrow\;\; & &\;\;\exp^{\times}\downarrow \ ARI(T)_{al} &\xrightarrow{\;\adari(\mathrm{pal})\;} &ARI(T)_{il} \end{array}$

This diagram encodes the passage of algebraic structure through the pal-twist and underpins combinatorial and dimensional claims regarding double shuffle relations (Komiyama, 2021).

7. Summary and Significance

The PAL-Set, in both algebraic and information-theoretic settings, encapsulates fundamental symmetries governing shuffle and stuffle behaviors, universality of code families, and bridges previously distinct structures via explicit automorphism. The rigidity and closure properties, as well as the analytic alignment criteria, provide robust tools for advancing the theory of MZVs and the performance and universality of channel coding frameworks. The PAL-Set stands as a central construct, enabling new insights into the combinatorics of multiple zeta values and the design and analysis of polar coding schemes.