Two-Branch Multiplicative-Coset Construction

• Two-Branch Multiplicative-Coset Construction is an algebraic framework using dual finite-field branches and coset certificates to enforce exact (J,L)-regularity and CSS orthogonality.
• It leverages explicit quotient-coset conditions and a normalized exhaustive search to prevent same-type 4-cycles and ensure consistent LDPC code performance.
• The construction integrates a two-stage design approach, combining base matrix design with cyclic lifting to optimize decoding and support finite-length performance evaluation.

Two-Branch Multiplicative-Coset Construction denotes an algebraic design paradigm in which incidence structures are generated from coset data arranged in two branch-indexed families. In the finite-field formulation developed for regular CSS LDPC base matrices, the construction uses two branches over a finite field to enforce exact $(J,L)$-regularity, CSS orthogonality, and same-type 4-cycle exclusion through explicit quotient-coset conditions (Okada et al., 22 May 2026). In a distinct earlier context, multiplication on double cosets of $GL(\infty)$ over a finite field is defined by a stabilization procedure with a canonical block permutation, yielding an associative semigroup on $P\backslash GL(\infty)/P$; this provides a closely related multiplicative-coset mechanism in an infinite-dimensional representation-theoretic setting (Neretin, 2013).

1. Finite-field branch architecture

The finite-field construction begins with a finite field $F$ and a multiplicative subgroup $M\le F^\times$. It uses two branches indexed by

$\lambda\in\{0,1\},$

with branchwise coefficient arrays

$$a_0^{(\lambda)},\ldots,a_{J-1}^{(\lambda)}\in F,\qquad b_0^{(\lambda)},\ldots,b_{J-1}^{(\lambda)}\in F.$$

The column index set is

$(\lambda,t,h)\in \{0,1\}\times F\times M.$

The $X$-rows are indexed by $(i,r)$, with $GL(\infty)$0 and $GL(\infty)$1, and the $GL(\infty)$2-rows are indexed by $GL(\infty)$3, with $GL(\infty)$4 and $GL(\infty)$5 (Okada et al., 22 May 2026).

Incidence is defined multiplicatively. A column $GL(\infty)$6 has ones in $GL(\infty)$7 at

$GL(\infty)$8

and in $GL(\infty)$9 at

$P\backslash GL(\infty)/P$0

Each branch is therefore a translated copy of a multiplicative subgroup orbit, with $P\backslash GL(\infty)/P$1 as translation coordinate and $P\backslash GL(\infty)/P$2 as subgroup coordinate (Okada et al., 22 May 2026).

The role of the two branches is structural. The incidence pattern itself enforces regularity; CSS orthogonality is enforced by pairing the overlaps of branch $P\backslash GL(\infty)/P$3 and branch $P\backslash GL(\infty)/P$4; and same-type 4-cycle exclusion is enforced by making same-type difference cosets disjoint across the two branches. The row weight is controlled by the subgroup size,

$P\backslash GL(\infty)/P$5

so the construction naturally targets even row weight $P\backslash GL(\infty)/P$6 (Okada et al., 22 May 2026).

2. Quotient-coset certificates

The regularity theorem states that the construction is automatically $P\backslash GL(\infty)/P$7-regular with

$P\backslash GL(\infty)/P$8

Every column has weight $P\backslash GL(\infty)/P$9, and every row has weight $F$0 (Okada et al., 22 May 2026).

For CSS orthogonality, an $F$1-row $F$2 and a $F$3-row $F$4 share a branch-$F$5 column exactly when

$F$6

A sufficient certificate is given by the conditions

$F$7

together with the cross-branch coset equality

$F$8

Under these conditions, every $F$9-row/$M\le F^\times$0-row overlap occurs either in neither branch or in both branches, so the total binary inner product is even: $M\le F^\times$1

Same-type 4-cycle exclusion is handled analogously. For $M\le F^\times$2-rows, two rows $M\le F^\times$3 and $M\le F^\times$4 share a branch-$M\le F^\times$5 column iff

$M\le F^\times$6

A sufficient condition preventing two same-type rows from sharing two columns is the conjunction of nonzero same-type differences,

$M\le F^\times$7

and disjoint same-type cosets across branches,

$M\le F^\times$8

with the analogous condition for the $M\le F^\times$9-coefficients. This guarantees no same-type 4-cycles in either Tanner graph (Okada et al., 22 May 2026).

These conditions are recast through the quotient map

$\lambda\in\{0,1\},$0

Letting $\lambda\in\{0,1\},$1 and $\lambda\in\{0,1\},$2, the paper records the necessary conditions

$\lambda\in\{0,1\},$3

and, for $\lambda\in\{0,1\},$4,

$\lambda\in\{0,1\},$5

The coefficient feasibility certificate is then expressed as

$\lambda\in\{0,1\},$6

while

$\lambda\in\{0,1\},$7

and

$\lambda\in\{0,1\},$8

The paper explicitly characterizes these quotient-coset conditions as sufficient certificates; it does not claim they are necessary in full generality (Okada et al., 22 May 2026).

3. Normalized search and two-stage design

Coefficient selection is reduced to a finite normalized exhaustive search. Because translating all coefficients in one branch does not change within-branch differences, and multiplying all coefficients by a common nonzero field element preserves coset relations, one may normalize

$\lambda\in\{0,1\},$9

After fixing $$a_0^{(\lambda)},\ldots,a_{J-1}^{(\lambda)}\in F,\qquad b_0^{(\lambda)},\ldots,b_{J-1}^{(\lambda)}\in F.$$0, $$a_0^{(\lambda)},\ldots,a_{J-1}^{(\lambda)}\in F,\qquad b_0^{(\lambda)},\ldots,b_{J-1}^{(\lambda)}\in F.$$1, and $$a_0^{(\lambda)},\ldots,a_{J-1}^{(\lambda)}\in F,\qquad b_0^{(\lambda)},\ldots,b_{J-1}^{(\lambda)}\in F.$$2, the orthogonality constraints determine each $$a_0^{(\lambda)},\ldots,a_{J-1}^{(\lambda)}\in F,\qquad b_0^{(\lambda)},\ldots,b_{J-1}^{(\lambda)}\in F.$$3 via the finite intersection

$$a_0^{(\lambda)},\ldots,a_{J-1}^{(\lambda)}\in F,\qquad b_0^{(\lambda)},\ldots,b_{J-1}^{(\lambda)}\in F.$$4

after which the search checks the same-type disjointness conditions (Okada et al., 22 May 2026).

For fixed $$a_0^{(\lambda)},\ldots,a_{J-1}^{(\lambda)}\in F,\qquad b_0^{(\lambda)},\ldots,b_{J-1}^{(\lambda)}\in F.$$5, the deterministic search cost is stated as

$$a_0^{(\lambda)},\ldots,a_{J-1}^{(\lambda)}\in F,\qquad b_0^{(\lambda)},\ldots,b_{J-1}^{(\lambda)}\in F.$$6

with $$a_0^{(\lambda)},\ldots,a_{J-1}^{(\lambda)}\in F,\qquad b_0^{(\lambda)},\ldots,b_{J-1}^{(\lambda)}\in F.$$7. The exhaustive search ranges over fields and subgroups satisfying the algebraic constraints above, and the paper gives examples for many $$a_0^{(\lambda)},\ldots,a_{J-1}^{(\lambda)}\in F,\qquad b_0^{(\lambda)},\ldots,b_{J-1}^{(\lambda)}\in F.$$8 pairs, including $$a_0^{(\lambda)},\ldots,a_{J-1}^{(\lambda)}\in F,\qquad b_0^{(\lambda)},\ldots,b_{J-1}^{(\lambda)}\in F.$$9, $(\lambda,t,h)\in \{0,1\}\times F\times M.$0, $(\lambda,t,h)\in \{0,1\}\times F\times M.$1, $(\lambda,t,h)\in \{0,1\}\times F\times M.$2, $(\lambda,t,h)\in \{0,1\}\times F\times M.$3, $(\lambda,t,h)\in \{0,1\}\times F\times M.$4, $(\lambda,t,h)\in \{0,1\}\times F\times M.$5, $(\lambda,t,h)\in \{0,1\}\times F\times M.$6, $(\lambda,t,h)\in \{0,1\}\times F\times M.$7; $(\lambda,t,h)\in \{0,1\}\times F\times M.$8, $(\lambda,t,h)\in \{0,1\}\times F\times M.$9, $X$0, $X$1, $X$2, $X$3, $X$4, $X$5, $X$6, $X$7, $X$8, $X$9; and $(i,r)$0 (Okada et al., 22 May 2026).

The overall design is explicitly split into two stages. Stage 1 selects the finite field $(i,r)$1, the subgroup $(i,r)$2, and the coefficients $(i,r)$3, thereby fixing exact $(i,r)$4-regularity, CSS orthogonality, and the absence of same-type 4-cycles. Stage 2 performs a cyclic lift, replacing each nonzero entry by a $(i,r)$5 circulant permutation matrix $(i,r)$6 with exponent labels

$(i,r)$7

The lift stage is used to preserve orthogonality via zero congruence constraints, prevent same-type base 6-cycles from closing, and exclude a chosen low-weight logical-support orbit. The base therefore fixes the degree distribution and the first girth constraints, while the lift injects randomness subject to exact algebraic checks (Okada et al., 22 May 2026).

4. Detailed $(i,r)$8-regular instantiation

The detailed example uses the $(i,r)$9 base over $GL(\infty)$00 with

$GL(\infty)$01

This base has column weight $GL(\infty)$02, row weight $GL(\infty)$03, $GL(\infty)$04 same-type simple base 6-cycles, and base length $GL(\infty)$05 (Okada et al., 22 May 2026).

The selected lift factor is

$GL(\infty)$06

so the full lifted length is

$GL(\infty)$07

The lift labels are chosen so that all same-type base 6-cycles have nonzero signed CPM-exponent sum, and therefore none closes in the lift. The lifted same-type Tanner graphs consequently have girth at least

$GL(\infty)$08

Orthogonality is preserved after lifting by imposing, for every $GL(\infty)$09-row and $GL(\infty)$10-row sharing exactly two base columns $GL(\infty)$11,

$GL(\infty)$12

This preserves

$GL(\infty)$13

The paper also excludes a specific weight-16 $GL(\infty)$14-type support family. It is built from $GL(\infty)$15 base columns and the $GL(\infty)$16-point lift-coordinate subgroup

$GL(\infty)$17

so the lifted support has weight

$GL(\infty)$18

The seed supports are

$GL(\infty)$19

and

$GL(\infty)$20

Translations and subgroup scalings generate an orbit of $GL(\infty)$21 distinct supports after duplicates are removed, and all $GL(\infty)$22 orbit members are reported as excluded by inconsistent congruence systems (Okada et al., 22 May 2026).

For the resulting lifted CSS code, the effective parameters are

$GL(\infty)$23

with certified distance interval

$GL(\infty)$24

More explicitly, the paper states

$GL(\infty)$25

and explicit witnesses give

$GL(\infty)$26

Hence

$GL(\infty)$27

5. Decoding procedure and finite-length performance

The decoding study for the detailed $GL(\infty)$28 instance uses joint log-domain belief propagation on the CSS factor graph with

$GL(\infty)$29

together with a fallback zero-damping run if needed (Okada et al., 22 May 2026).

Post-processing is low-complexity and deterministic for small residual syndromes. The rules listed in the paper are local linear solve, prefix-size search, flip-history candidate set, path-based closure, common-column correction, syndrome-2 core repair, and small-residual exact/beam search for $GL(\infty)$30–$GL(\infty)$31 unsatisfied checks. This decoding stack is part of the paper’s explicit separation between combinatorial base design, algebraically constrained lifting, and finite-length decoding evaluation (Okada et al., 22 May 2026).

At depolarizing probability

$GL(\infty)$32

the reported measurements are $GL(\infty)$33 trials, $GL(\infty)$34 recorded failures, $GL(\infty)$35 corrected by post-processing, and final frame error rate

$GL(\infty)$36

The paper characterizes these as finite-length decoding data for the detailed example. It also notes a corresponding limitation: the decoding data are finite-sample performance measurements, not asymptotic threshold statements, and the exact distance of the lifted example is not proved (Okada et al., 22 May 2026).

6. Relation to double-coset semigroups, scope, and limitations

A related multiplicative-coset construction appears in the study of $GL(\infty)$37, the group of all infinite matrices $GL(\infty)$38 over a field $GL(\infty)$39 such that $GL(\infty)$40 has only finitely many nonzero entries. Viewing this as a block-matrix group of size

$GL(\infty)$41

one considers the parabolic subgroup $GL(\infty)$42 corresponding to that partition and the subgroup

$GL(\infty)$43

where the natural projection extracts the middle $GL(\infty)$44 block. The objects of interest are the double cosets

$GL(\infty)$45

with equivalence relation

$GL(\infty)$46

Multiplication is defined by a shift-and-stabilize rule: representatives are placed in a sufficiently large block decomposition, a fixed “intertwining” matrix $GL(\infty)$47 exchanges the two $GL(\infty)$48-blocks in the middle, and the product is taken in stabilized form

$GL(\infty)$49

The paper proves that this operation is independent of representatives and of the choice of sufficiently large $GL(\infty)$50, and states as Theorem 1.4 that

$GL(\infty)$51

is a well-defined associative operation. It also defines an involution $GL(\infty)$52 by inversion of representatives and gives

$GL(\infty)$53

as Proposition 1.5. On the representation-theoretic side, if $GL(\infty)$54 is a unitary representation of $GL(\infty)$55 and $GL(\infty)$56 is the subspace of $GL(\infty)$57-fixed vectors, then for $GL(\infty)$58 with representative $GL(\infty)$59,

$GL(\infty)$60

and

$GL(\infty)$61

on $GL(\infty)$62 (Neretin, 2013).

The 2013 paper does not name its construction “two-branch,” but it explicitly describes the procedure as very similar in spirit to colligations or transfer-matrix multiplication, and its stabilized block decomposition is branch-like in the sense that two large auxiliary $GL(\infty)$63-blocks are inserted and then exchanged by a canonical bridge matrix. This suggests a common abstract schema across the two papers: stabilization into parallel branches, a canonical swap or orbit-matching device, and quotient invariance under an equivalence relation.

Within the CSS LDPC setting, the method is presented as a fully explicit finite-field construction of regular CSS LDPC bases whose regularity, orthogonality, and same-type 4-cycle exclusion reduce to checkable coset conditions. It works for many $GL(\infty)$64 pairs, separates base design from lift-label design and decoding evaluation, and uses a normalized search intended to be reproducible. At the same time, the quotient-coset conditions are stated only as sufficient, not fully necessary; the $GL(\infty)$65 example is certified only within

$GL(\infty)$66

exclusion of one weight-16 orbit is not exclusion of all weight-16 logical operators; and the paper describes the framework as field-independent in spirit, suggesting extensions to finite groups, rings, or modules with quotient-coset conditions replaced by orbit-equality or orbit-disjointness tests (Okada et al., 22 May 2026).