Dyadic Ordered Template Basis

• Dyadic ordered template basis is an orbit-decomposition framework that organizes low- and intermediate-weight codewords of decreasing monomial codes by converting weights into canonical dyadic signatures.
• It delineates four template families—disjoint sums, nested-block kernels, complementary flips, and shared 3-term kernels—to classify codewords and compute exact multiplicities via inclusion–exclusion and LTA group actions.
• The framework refines the weight hierarchy in polar codes by using dyadic ordering, thereby linking local kernel patterns with global algebraic structure and enhancing code spectrum enumeration.

Dyadic ordered template basis is the organization of low- and intermediate-weight codewords of a decreasing monomial code $C(I)$ by two coupled structures: the dyadic expansion of the normalized Hamming weight and a finite set of canonical residual-kernel templates. In "Generalized Weight Structure of Polar Codes: Selected Template Polynomials" (Rowshan et al., 15 Jan 2026), polar codes are treated as decreasing monomial codes whose algebraic structure is governed by the lower-triangular affine (LTA) group. Within that framework, the dyadic-ordered template basis consists of four template families—disjoint sums, nested-block kernels, complementary flips, and shared 3-term kernels—together with admissible monomial heads and their LTA orbits. The term "basis" is used in the sense of an orbit-decomposition rather than a linear basis, and the construction is intended to characterize and enumerate codewords in the low and intermediate weight regime.

1. Canonical dyadic form of normalized weight

The framework begins with a factorization

$P(x)=h(x)\sum_{i=1}^q f_i(x),$

where $h$ is a monomial of degree $\deg(h)=r-a_{\max}$, each $f_i$ is a monomial of degree $\le a_{\max}$, and $\deg(P)=r$ (Rowshan et al., 15 Jan 2026). Setting

$$d=2^{m-r}, \qquad \Sigma(P)\triangleq \frac{wt(\mathrm{ev}(P))}{d},$$

the weight computation is reduced to the residual sum $F=\sum_i f_i$. By inclusion–exclusion,

$$wt(P)=d\,\Sigma(F), \qquad \Sigma(F)=\sum_{\varnothing\neq S\subseteq\{1,\dots,q\}}(-2)^{|S|-1}\,2^{a_{\max}-u_S},$$

where

$P(x)=h(x)\sum_{i=1}^q f_i(x),$0

The decisive structural point is that the normalized weight is always dyadic. If

$P(x)=h(x)\sum_{i=1}^q f_i(x),$1

then

$P(x)=h(x)\sum_{i=1}^q f_i(x),$2

Equivalently, every normalized weight admits a unique binary expansion

$P(x)=h(x)\sum_{i=1}^q f_i(x),$3

This dyadic decomposition is the ordering device for the entire construction. It converts the Hamming weight of a monomial-code codeword into a canonical bit pattern, thereby making weight comparison an algebraic comparison of dyadic signatures rather than an ad hoc case analysis.

2. Structural templates

For fixed ambient degree $P(x)=h(x)\sum_{i=1}^q f_i(x),$4, the paper identifies four canonical residual-kernel templates $P(x)=h(x)\sum_{i=1}^q f_i(x),$5, possibly multiplied by a disjoint head $P(x)=h(x)\sum_{i=1}^q f_i(x),$6, whose normalized weights lie in the low or intermediate range and which generate all such codewords up to LTA action (Rowshan et al., 15 Jan 2026).

Type	Kernel form	Normalized weight
I	$P(x)=h(x)\sum_{i=1}^q f_i(x),$7, $P(x)=h(x)\sum_{i=1}^q f_i(x),$8, disjoint supports	$P(x)=h(x)\sum_{i=1}^q f_i(x),$9
II	$h$0, $h$1, $h$2	$h$3
III	$h$4, $h$5, $h$6	$h$7
IV	$h$8 or $h$9, shared 3-term cubic kernels	$\deg(h)=r-a_{\max}$0

The Type I disjoint $\deg(h)=r-a_{\max}$1-sum template is

$\deg(h)=r-a_{\max}$2

with $\deg(h)=r-a_{\max}$3 and $\deg(h)=r-a_{\max}$4 for $\deg(h)=r-a_{\max}$5. Its weight is

$\deg(h)=r-a_{\max}$6

The Type II nested-block, or degree-drop, kernel is

$\deg(h)=r-a_{\max}$7

where $\deg(h)=r-a_{\max}$8, $\deg(h)=r-a_{\max}$9, all supports are disjoint except that each $f_i$0, and $f_i$1.

The Type III complementary-flip template is

$f_i$2

with $f_i$3, $f_i$4, and

$f_i$5

Its full weight expression is

$f_i$6

The Type IV shared 3-term cubic kernels are

$f_i$7

and

$f_i$8

Each has $f_i$9, hence

$\le a_{\max}$0

These kernels may be nested under an arbitrary disjoint head $\le a_{\max}$1 of degree $\le a_{\max}$2.

3. Dyadic signatures and the ordering principle

To each template $\le a_{\max}$3 the construction assigns a dyadic signature through the binary expansion

$\le a_{\max}$4

The ordering is lexicographic on the bit-vectors $\le a_{\max}$5, with the equivalent criterion that $\le a_{\max}$6 precedes $\le a_{\max}$7 if $\le a_{\max}$8 (Rowshan et al., 15 Jan 2026).

For normalized weights below $\le a_{\max}$9, the source gives the rough basis hierarchy

$\deg(P)=r$0

and refers to templates whose signatures begin $\deg(P)=r$1, $\deg(P)=r$2, $\deg(P)=r$3, and so on.

The significance of this ordering is that it ranks template families by the dyadic content of their normalized weights rather than only by the scalar value of the weight. This suggests a refined hierarchy inside the low-weight spectrum: codewords are not merely grouped by $\deg(P)=r$4, but by the binary structure of $\deg(P)=r$5, which is the invariant naturally produced by the inclusion–exclusion formula.

4. Completeness in the low and intermediate regime

The central structural theorem is a spanning statement for codewords of sufficiently small weight. Every codeword $\deg(P)=r$6 with

$\deg(P)=r$7

can be written, up to an LTA automorphism, in the form

$\deg(P)=r$8

where $\deg(P)=r$9 is a monomial head and $$d=2^{m-r}, \qquad \Sigma(P)\triangleq \frac{wt(\mathrm{ev}(P))}{d},$$0 is one of the four template types, possibly nested under $$d=2^{m-r}, \qquad \Sigma(P)\triangleq \frac{wt(\mathrm{ev}(P))}{d},$$1 (Rowshan et al., 15 Jan 2026).

The theorem is refined by weight level. Any codeword with $$d=2^{m-r}, \qquad \Sigma(P)\triangleq \frac{wt(\mathrm{ev}(P))}{d},$$2 arises either from a disjoint-2-sum $$d=2^{m-r}, \qquad \Sigma(P)\triangleq \frac{wt(\mathrm{ev}(P))}{d},$$3 or from a complementary flip $$d=2^{m-r}, \qquad \Sigma(P)\triangleq \frac{wt(\mathrm{ev}(P))}{d},$$4 with $$d=2^{m-r}, \qquad \Sigma(P)\triangleq \frac{wt(\mathrm{ev}(P))}{d},$$5. Any codeword with $$d=2^{m-r}, \qquad \Sigma(P)\triangleq \frac{wt(\mathrm{ev}(P))}{d},$$6 arises from a nested degree-drop kernel $$d=2^{m-r}, \qquad \Sigma(P)\triangleq \frac{wt(\mathrm{ev}(P))}{d},$$7 with $$d=2^{m-r}, \qquad \Sigma(P)\triangleq \frac{wt(\mathrm{ev}(P))}{d},$$8 or $$d=2^{m-r}, \qquad \Sigma(P)\triangleq \frac{wt(\mathrm{ev}(P))}{d},$$9, or from a shared 3-term kernel $F=\sum_i f_i$0 or $F=\sum_i f_i$1, possibly multiplied by a disjoint head $F=\sum_i f_i$2. The next possible weight,

$F=\sum_i f_i$3

arises uniquely from the disjoint-3-sum of three cubics $F=\sum_i f_i$4 with $F=\sum_i f_i$5.

The paper also gives weight-$F=\sum_i f_i$6 examples of the form

$F=\sum_i f_i$7

and

$F=\sum_i f_i$8

matching Type I with $F=\sum_i f_i$9 or Type III with $$wt(P)=d\,\Sigma(F), \qquad \Sigma(F)=\sum_{\varnothing\neq S\subseteq\{1,\dots,q\}}(-2)^{|S|-1}\,2^{a_{\max}-u_S},$$0.

The resulting family is

$$wt(P)=d\,\Sigma(F), \qquad \Sigma(F)=\sum_{\varnothing\neq S\subseteq\{1,\dots,q\}}(-2)^{|S|-1}\,2^{a_{\max}-u_S},$$1

with all admissible heads $$wt(P)=d\,\Sigma(F), \qquad \Sigma(F)=\sum_{\varnothing\neq S\subseteq\{1,\dots,q\}}(-2)^{|S|-1}\,2^{a_{\max}-u_S},$$2 and all LTA-inequivalent kernels $$wt(P)=d\,\Sigma(F), \qquad \Sigma(F)=\sum_{\varnothing\neq S\subseteq\{1,\dots,q\}}(-2)^{|S|-1}\,2^{a_{\max}-u_S},$$3. The source states that $$wt(P)=d\,\Sigma(F), \qquad \Sigma(F)=\sum_{\varnothing\neq S\subseteq\{1,\dots,q\}}(-2)^{|S|-1}\,2^{a_{\max}-u_S},$$4 forms a basis in the sense of an orbit-decomposition for all codewords whose weight lies in the low or intermediate region. A common misunderstanding would be to read this as a statement about a linear basis of $$wt(P)=d\,\Sigma(F), \qquad \Sigma(F)=\sum_{\varnothing\neq S\subseteq\{1,\dots,q\}}(-2)^{|S|-1}\,2^{a_{\max}-u_S},$$5; the formulation in the source is narrower and explicitly orbit-theoretic.

5. LTA action and orbit multiplicities

The automorphism mechanism is the lower-triangular affine group

$$wt(P)=d\,\Sigma(F), \qquad \Sigma(F)=\sum_{\varnothing\neq S\subseteq\{1,\dots,q\}}(-2)^{|S|-1}\,2^{a_{\max}-u_S},$$6

which acts on variables by

$$wt(P)=d\,\Sigma(F), \qquad \Sigma(F)=\sum_{\varnothing\neq S\subseteq\{1,\dots,q\}}(-2)^{|S|-1}\,2^{a_{\max}-u_S},$$7

and on a monomial $$wt(P)=d\,\Sigma(F), \qquad \Sigma(F)=\sum_{\varnothing\neq S\subseteq\{1,\dots,q\}}(-2)^{|S|-1}\,2^{a_{\max}-u_S},$$8 by

$$wt(P)=d\,\Sigma(F), \qquad \Sigma(F)=\sum_{\varnothing\neq S\subseteq\{1,\dots,q\}}(-2)^{|S|-1}\,2^{a_{\max}-u_S},$$9

This action extends multiplicatively to any polynomial $P(x)=h(x)\sum_{i=1}^q f_i(x),$00 (Rowshan et al., 15 Jan 2026).

For a seed polynomial $P(x)=h(x)\sum_{i=1}^q f_i(x),$01, the orbit size is determined by head and kernel stabilizers. The stabilizer of a head $P(x)=h(x)\sum_{i=1}^q f_i(x),$02 has size

$P(x)=h(x)\sum_{i=1}^q f_i(x),$03

The stabilizer of a kernel $P(x)=h(x)\sum_{i=1}^q f_i(x),$04, viewed as a sum of tails $P(x)=h(x)\sum_{i=1}^q f_i(x),$05, contributes an additional exponent $P(x)=h(x)\sum_{i=1}^q f_i(x),$06 of free LTA parameters, while collisions among tails subtract an exponent $P(x)=h(x)\sum_{i=1}^q f_i(x),$07. The master formula is

$P(x)=h(x)\sum_{i=1}^q f_i(x),$08

This formula turns the template basis into an exact counting mechanism. Since distinct templates, up to head and kernel equivalence, have disjoint orbits, summing the orbit sizes over the allowed $P(x)=h(x)\sum_{i=1}^q f_i(x),$09 recovers the exact multiplicity of codewords at the corresponding template weight.

6. Interpretation within the algebraic weight structure of polar codes

The dyadic-ordered template basis is presented as part of a generalized algebraic description of the weight structure of polar codes, viewed through decreasing monomial codes and LTA symmetry (Rowshan et al., 15 Jan 2026). Its two defining moves are, first, to express the normalized weight of any residual sum in canonical dyadic form and, second, to isolate four minimal kernel patterns whose normalized weights occupy successive dyadic thresholds such as $P(x)=h(x)\sum_{i=1}^q f_i(x),$10, $P(x)=h(x)\sum_{i=1}^q f_i(x),$11, $P(x)=h(x)\sum_{i=1}^q f_i(x),$12, and $P(x)=h(x)\sum_{i=1}^q f_i(x),$13.

In this formulation, low- and intermediate-weight codewords are not treated as a miscellaneous collection of exceptional cases. They are generated from a small number of kernels, multiplied by admissible heads, and then expanded into complete LTA orbits. The paper’s conclusion is that this basis is closed under LTA action, yields full orbits and multiplicities, and is complete for all low- and intermediate-weight codewords of any decreasing monomial code.

The explicit scope of the structural theorem is the regime $P(x)=h(x)\sum_{i=1}^q f_i(x),$14. A plausible implication is that the dyadic ordering provides a scalable interface between local kernel classification and global spectrum enumeration: inclusion–exclusion determines the dyadic signature, the template family determines the structural type, and the LTA orbit formula determines multiplicity.